n julio 3_takacs - montmort problem

7/27/2019 N Julio 3_Takacs - Montmort Problem

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The Problem of CoincidencesLAJOS TAKAC S

Communicated by B.L . VAN DERWAERDEN

SummaryThis paper dea ls wi th the or ig in , h is tory and var ious appearances of the

proble m of co incidences (matches , r encontres) in the theory of probabi l i ty .

1. IntroductionThe problem of co inc idences (matches , r encontres) or ig ina tes in the game of

thir teen ( jeu du tre ize) and was proposed in 1708 by PIERRE R~MOND DEMONTMORT (1678-1719) in his bo o k [-38 p. 185]. I n a letter to JO HAN NBERNOULLI (1667-1748), da ted N ov em be r 15, 17 10, MO NTM ORT gave th esolu t ion of the problem which he propose d in 1708. He added tha t he had foundthe genera l so lu t ion of the game of th i r teen , but i t would be too long to g ive a llthe de ta i l s . Ac tua l ly , he d id not even wr i te down the genera l so lu t ion . Thegen eral s olu tion was given by NIKOLAUS BERNOULLI (1687-1759), a ne phe w ofJOHANN BERNOULLI, n a letter to MONTMORT,da ted Feb ruary 26, 1711. Bothle t ters are repr inted in the second edit ion of MONTMORT's book [38] . See [393pp. 303-307 and pp. 308-314 respectively. MONTMORT's problem had greatinf luence on the deve lopm ent of probabi l i ty theory , and the a im of th is paper i sto give a histor ical acco unt of the result o f this inf luence.

2. The Game of ThirteenThe game of th i r teen ( jeu du t reize) i s p layed by any nu mb er of p layers wi th

a Fren ch deck of 52 cards which consists of an ace (1) , nine nu me rals (2 to 10) ,kna ve (11) , queen (12) and king (13) in each of four suits (spades, hear ts ,diam ond s, and c lubs). The ran k o f each card is indicate d in brackets . F irst , theplayers choose a banker . The ban ker shuff les the ca rds and turns up th i r teencards on e af ter the other . As he turns up the cards he cal ls ou t the nam es of thecards in a suit in orde r o f ran k (ace , two, . . . , king) . I f in this sequence n o cardcoinc ides wi th the na me of the ca rd ca lled, the banker pays each one of the

Archive for History of Ex ac t Sciences, Volume 21, by Springer-Verlag 1980


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230 L. TAK,~CSp l a y e r s a n d y i e ld s t h e b a n k t o t h e p l a y e r o n h i s r i g h t. B u t i f t h e r e i s ac o i n c i d e n c e i n t h e t u r n i n g o f t h e t h i r t e e n c a r d s , fo r e x a m p l e , if a n a c e t u r n s u pa t t h e f i r s t s t e p w h e n t h e b a n k e r h a s c a l l e d a c e , o r a t w o a t t h e s e c o n d s t e p w h e nt h e b a n k e r h a s c a l l e d t w o , t h e b a n k e r t a k e s a l l t h e s t a k e s a n d b e g i n s a g a i n a sb e f o r e .

I n 1 7 0 8 a t t h e e n d o f h i s b o o k M O N T M O R T [ -3 8 , p p . 1 8 5 - 1 8 9 ] , [ 3 9 , p . 2 7 8 ]p r o p o s e d f o u r p r o b l e m s f o r s o l u t i o n . T h e f i r s t p r o b l e m w a s t o d e t e r m i n e t h ep r o b a b i l i t y t h a t t h e b a n k e r w i n s a g a m e i n t h e g a m e o f t h ir te e n .

M o r e g e n e r a l l y , i t c a n b e a s k e d w h a t i s t h e p r o b a b i l i t y t h a t t h e b a n k e r w i n sa g a m e i f i n s t e a d o f 1 3 t h e r e a r e n c a r d s i n a s u i t a n d i f i n s t e a d o f 4 t h e r e a r e ss u i t s i n t h e d e c k , a n d i n e a c h g a m e t h e b a n k e r n a m e s n c a r d s i n a s u i t .

I t i s c o n v e n i e n t t o s t u d y t h e t w o c a s e s , s = 1 a n d s _->1, s e p a r a t e l y a n d t o u s et h e f o l l o w i n g m o d e l s :

M o d e l I . A box con t a i ns n cards m arked 1 , 2 . . . . . n, and a l l the n cards are drawnf r o m t h e b o x o n e b y o n e w i t h o u t r e p la c e m e n t . E v e r y o u t c o m e o f t h is r a n d o m t r ia lc a n b e r e p r es e n te d b y a p e r m u t a t io n o f 1 , 2 , . . . , n . I t is a s s u m e d t h a t a l l t h e n !perm ut a t i ons o f 1 , 2 , . . . , n a re eq ua l l y p robab l e .

W e s a y t h a t a c o i n c i d e n c e o c c u r s a t t h e i n d r a w i n g i f t h e c a r d d r a w n ism a r k e d i ( i = 1 , 2 , . . ., n ). D e n o t e b y P ( n , k ) t h e p r o b a b i l i t y t h a t i n t h e n d r a w i n g sw e h a v e a t l e a s t k c o i n c i d e n c e s ( k = 0, 1 , . . . , n ). W e w r i t e(1 ) P ( n , 1 ) = D ( n ) / n ! ,w h e r e D ( n ) d e n o t e s t h e n u m b e r o f p e r m u t a t i o n s o f 1, 2, . . . , n in w h i c h a t l e a sto n e c o i n c i d e n c e o c c u r s , a n d(2) 1 - P ( n , 1) = Q(n ) /n !,w h e r e Q ( n ) d e n o t e s t h e n u m b e r o f p e r m u t a t i o n s o f 1, 2 , . . . , n in w h i c h n o c o i nc i -d e n c e o c c u r s .

Mode l I I . A bo x c on t a i ns s s e t s o f cards, each s e t cons i s t ing o f n cards m arke d1 , 2 , . . . , n . Fr om t he box n cards a re d rawn one b y one wi t ho u t r ep l acem en t . Thereare n s ( n s - 1 ) . . . ( n s - n + 1 ) p o s s i b l e ou t com es and t he se a re a s sum ed t o be equa l l yprobab l e .

W e s a y a g a i n t h a t a c o i n c i d e n c e o c c u r s a t t h e i h d r a w i n g i f t h e c a r d d r a w n i sm a r k e d i ( i = 1 , 2 , . . . , n ) . D e n o t e b y P ( n , s , k ) t h e p r o b a b i l i t y t h a t i n t h e nd r a w i n g s w e h a v e a t l e a s t k c o in c i d e n c e s ( k = 0 , 1 , . . . , n ) . O f c o u r s e P( n, 1, k)= P ( n , k ) fo r O


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The Problem of Coinc idences 231I n w h a t f o l l o w s w e s h a l l s u m m a r i z e f i r s t t h e b a s i c r e s u l t s f o r P ( n , k ) a n dP( n , s , k ) a n d t h e n w e s h a l l d i s c u s s t h e i r h i s t o r i c a l d e v e l o p m e n t .

3. Basic R esults for M odel ID e f i n e A i ( i = 1 , 2 . . . . , n ) a s t h e e v e n t t h a t a c o i n c i d e n c e o c c u r s a t t h e i t h

d r a w i n g . D e n o t e b y 4 , t h e n u m b e r o f c o i n c i d e n c e s i n th e n d r a w i n g s a n d l e t 40~ 0 .

W e s h a l l g i v e t w o m e t h o d s f o r f i n d i n g t h e d i s t r i b u t i o n o f 4~ f o r n = 1, 2 , . . . .O n e i s b a s e d o n a r e c u r r e n c e f o r m u l a , a n d t h e o t h e r o n a g e n e r a l t h e o r e m o fp r o b a b i l i t y .

O b v i o u s l y(3) P { ~ , = k } = ( ~ ) Q ( n - k ) / n !w her e Q ( n) is d e f i ne d b y ( 2) f o r n => 1 an d Q ( 0) = 1 . Th us t he p r ob l em o f f i nd i ngt h e d i s t r i b u t i o n o f 4, c a n b e r e d u c e d t o t h e p r o b l e m o f fi n d in g Q ( j) f o r j= 0 , 1 , . . . , n . S i n c e(4) ~ P { 4 . = k } = 1 ,

k = 0

i t f o l l ows f r om ( 3 ) t ha tn n "3-, Q ( - j )(5) j~oj,.~-_~ )! 1

f o r n = 0 , 1 , 2 , . . . . F o r m u l a ( 5) is a lr e a d y a r e c u rr e n c e fo r m u l a f o r t h e d e-t e r m i n a t i o n o f Q ( n ) ( n = 0 , 1 , 2 , . . . ) . H o w e v e r , i t is e a s y t o f in d a n e x p l ic i te x p r e s s i o n f o r Q(n) . M u l t i p l y i n g (5 ) b y x " a n d s u m m i n g f o r n = 0 , 1 ,2 , . . , , w e g e t(6 )for Ix l < 1 . H en ce(7 )

Q ( n ~ ) ) x " = e - X ( 1 - x ) - in = 0 F / !

Q ( n)= ~ ( - 1 /n! j = o J !

f o r n = 0 , 1, 2 , . . . . T h e s a m e r e s u lt c a n a l so b e o b t a i n e d b y u s i ng t h e r e c u r r e n c ef o r m u l a(S) Q (n) -- (n - 1) [Q (n - 1) + Q (n - 2)]f or n > 2 w h e r e Q ( 0 ) = I a n d (2 ( 1) = 0.

B y a g e n e r a l t h e o r e m o f C . J O RD A N [ 2 4 ] i f A 1 , A 2 . .. . A , a r e r a n d o m e v e n t sa n d 4 , d e n o t e s t h e n u m b e r o f e v e n ts o c c u r r i n g a m o n g A 1 , A z . . . . , A , , t h e n


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232 L. TAKACS

r=k

f o r k = 0 , 1 , . . . , n a n d(10) P { ~ , > k } = ~ ( - 1 ) r - k ( r - l ) B r~ = k k - 1f o r k = 1 , 2 . . . . , n w h e r e B o = 1 a n d(11) B~= ~ P{A ,1Ai2. ..A,r }

l


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The Problem of Coincidences 233I n t h i s c a s e

S r

( 1 8 ) r~~ i i l A i2 "'" A i r} - n s ( n s - 1) . . . ( ns - r + 1)f o r r = 1, 2 , . . . , n a n d b y (9 ) w e o b t a i n t h a t

1 & ( - 1 ) r - k n s ( n s - s ) . .. ( n s - r s + s)(1 9) P { ~ , ( s ) = k } = ~ - . Lf or k = 0 , 1 , . . . , n a n d b y (10)

1 " ( 1 ) r - k n s ( n s - - s ) . . . ( n s - - r s + s )

f o r k = 1 , 2 , . . . , n . B y ( 1 9 ) w e o b t a i n t h a t(21) l im P {~,(s) = k} = e - 1/k!f or k = 0 , 1 , 2 , . . , a n d fo r a n y s = 1 , 2 , . . . .

5 . M o n t m o r t ' s R e s u l ts

F i r s t , i n 1 7 0 8 M O N T M O R T [ 3 8 , p p . 5 4 - 6 4 ] d i s c u s s e d t h e p r o b l e m o f f i n d i n gt h e p r o b a b i l i ty(22) P ( n , 1 = D ( n ) / n !f o r M o d e l I a n d f o r n = 1 3. F o r b r e v i t y , w e s h a ll w r i t e(23) P * ( n ) = D ( n ) / n !t ha t i s , P * ( n ) i s t h e p r o b a b i l i t y t h a t a t l e a s t o n e c o i n c i d e n c e o c c u r s i n t h e t r i a ld e s c r i b e d i n M o d e l I .

W i t h o u t g i v in g a n y j u s ti f i c a ti o n M O N T M O RT s t a t e d t h a t( n - 1) P * ( n - 1) + P * ( n - 2 )(24) P * ( n ) n

f or n = 2 , 3 . . . . w h e r e P * ( 0 ) = 0 a n d P * ( 1 ) = I . B y u si n g t h e re c u r re n c e fo r m u l a(24) MONTMORT ca lcu la t ed P * ( n ) f o r n < 1 3 a n d o b t a i n e d t h a t(25) P*( 13)-= 109339663172972800"F u r t h e r m o r e , M O N T M O R T s t a t e d t h a t(26) P * ( n ) = ( - 1 ) i - 1i=1 i!


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234 L. TAKACSf o r n > 1 , a n d c o n c l u d e d t h a t(27) l im P* (n) = 1 - e - 1.S e e a l s o M O N T M O R T [ 3 9 , p p . 1 3 0 - 1 4 3 ] .

M O N T M O R T 'S r e s u l t (2 4) a r o u s e d t h e i n t e r e s t o f s e v e r a l m a t h e m a t i c i a n s a n dn o w w e h a v e v a r i o u s p r o o f s f o r (2 4). F o r m u l a (2 6) fo l l o w s e a s i ly f r o m ( 24 ). I f w ew r i t e ( 2 4 ) i n t h e f o r m o f(28) P * ( n ) - P * ( n - 1 ) = - [ P * ( n - 1 ) - P * ( n - 2 )] /nw h e r e n > 2 a n d a p p l y (2 8) r e p e a t e d l y , t h e n w e o b t a i n(29) P * (n) - P * (n - 1) = ( - 1)" - 1 I n !f o r n > 1 a n d t h i s i m p l i e s ( 26 ). T h e l i m i t (2 7) is a n i m m e d i a t e c o n s e q u e n c e o f (2 6).

O n e w o n d e r s h o w M O N T M O R T a r r i v e d a t (2 4). I t se e m s l i k e ly t h a t h eo b t a i n e d ( 2 4 ) e m p i r i c a l l y .

T h e f ir st e d i t i o n o f M O N T M O R T ' S b o o k [ 3 8 ] is d i s cu s s e d b r i e f l y b yF . N . D A V lD [ 8 , p p . 1 4 0 - 1 6 0 3 , a n d t h e s e c o n d e d i t i o n [ 3 9 ] b y 1 . T O D H U N T ER[ 50 , pp . 78 - 13 4] . S ee , i n pa r t i cu l a r , pp . 91 - 93 , 105, 115 , 116, 120 - 121 , 122 - 1 25 i n[ 5 0 ] . S e e a l s o K . JO R D A N [ 2 5 , p p . 4 3 1 - 4 4 1 ] , [ 2 6 , p p . 4 2 9 - 4 3 9 ] .

6. Johann Bernoul l i ' s Contribut ion

M O N T M O R T s e n t a c o p y o f h i s b o o k [ 3 8 ] t o J O H A N N B E R N O U L LI f o rp e r u s a l . I n h i s l e t t e r t o M O N T M O R T , d a t e d M a r c h 1 7 , 1 7 1 0 , B E R N O U L L Ic o m m e n t e d o n t h e b o o k a n d h i s l e t t e r i s r e p r i n t e d i n t h e s e c o n d e d i t i o n o fM O N T M O R T 'S b o o k [ 3 9 , p p . 2 8 3 - 2 9 8 ] p u b l i s h e d i n 1 71 3.

J O H A NN B E R N O U LL I b e l i e v e d t h a t t h e s o l u t i o n o f M O N T M O R T'S p r o b l e mw a s i m p o s s i b l e b e c a u s e t h e l e n g t h y c a l c u l a t i o n s n e e d e d t o f in d P ( 1 3 , 4 , 1) c o u l d n o tb e f i n i s h e d i n a l i f e t i m e [ 3 9 , p . 2 9 8 ] . H e r e m a r k e d [ 3 9 , p . 2 9 0 ] t h a tM O N T M O R T ' s f o r m u l a ( 2 6 ) , w h i c h i s b e a u t i f u l a n d c u r i o u s , c a n e a s i l y b e o b -t a i n e d f r o m (2 4). I n d e e d i f w e w r i t e ( 2 4) i n t h e f o r m o f (2 8), t h e n w e g e t (2 6)eas i ly .

I n h i s l e t t e r , J O H A N N B E R N O U L L I [ 3 9 , p . 2 9 0 ] m e n t i o n e d a l s o t h a t

( 3 0 ) " 1 " P * ( n - j )

f o r n > l . T h i s e qu a t i on i s t r i v i a l l y t r ue . I f w e pu t ( 3 ) i n t o ( 4) , t he n w e ge t (30 ). I ts e e m s t h a t B E R N O U L L I d i d n o t n o t i c e t h a t ( 30 ) i s o b v i o u s l y tr u e , a n d t h a t i ti m p l i e s M O N T M O R T ' s f o r m u l a ( 2 6 ) .I n h i s r e p l y t o BERNOULLI,d a t e d N o v e m b e r 1 5 , 1 71 0, M O N T M O R T [ 3 9 , p p .3 0 3 - 3 0 7 ] w r o t e t h a t h e h a d f o u n d t h e g e n e r al s o l u t io n o f t h e g a m e o f t h i rt e e na n d t h a t b y h i s c a l c u l a t i o n s [ 3 9 , p . 3 0 4 ]


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Th e Pro blem of Coincidences 23569056823706869897

(31) 2P( 13, 4, 1 ) - 1 - 241347817621535625"Actual ly, MONTMORT made a s l ight error in calcula t ing (31) . For a correct ionsee N . BERNOULLI [39, p. 32 4].

7. Nikolaus Bernoull i 's ContributionAlong with JOHANN BERNOULLI'S comments, his nephew. NIKOLAUS BER-

NOULLI a lso sent his remarks to MONTMORT. These remarks are reprinted inthe se con d ed i t ion MONTMORT'S bo ok [39, pp. 2 99-30 3] . N. BERNOULLI gavetwo proofs for MONTMORT's formula (26) . Both proofs a re remarkable andde m ons t r a t e t he i nge nu i ty o f t he i r i nve n to r .N. BERNOULLI [-39, p. 301 ] gav e th e follo win g p ro of fo r (26). In (26) we c anwr i t e

Z(32) D( n) = L D( n , j )d= lw he r e D( n , j ) i s the n um ber of pe r mu ta t ion s of 1 ,2 . . . , n in which the re i s aco inc iden ce a t the j th p lace and the re i s no co inc idence be fore . He dem ons t ra tedtha t(33) D ( n , j ) = ~ ( - 1 ) i ( n - i - i ) !

i =

for j = I , 2 , . . . , n . F ro m (33) he conc lu ded tha t(34) D ( n , j ) = ~ ( - 1 ) z m ( n - l - i ) tj = l i= o i + 1for rn= 1 ,2 . . . , n . I f m = n in (34), we get D( n ) and (26) follows from (23).

F o r m ul a (33) c a n a l so be p r ove d by u s ing t he r e c u r r e nc e f o r mu l a(35) D (n, j + 1) = D (n,j) - D (n - 1,j)f or n = 2 , 3 . . . . a n d j = l , 2 . . . . , n - 1 w h er e D ( n , 1 ) = ( n - 1 ) ! f or n = l , 2 , . . . . T oobta in (35), we d iv ide the se t o f pe rm uta t io ns of 1 ,2 . . . . , n in which the re i s nocoinc idence a t the 2ha, 3rd, . . ., j th places and there is a co incide nce a t the ( j+ 1)Stplace , into two dis joint subsets such that in the f i rs t se t the f i rs t e lement i s 1 andin the sec on d set the first ele m en t is 4= 1.

In his seco nd p ro of of (26) N. BERNOULLI [39, pp. 30 1-30 2] de m on stra tedt ha t(36) P * ( n ) = 1 -4 ( n - 1) [ D ( n - 1 ) - ( n - 2 ) ! + D ( n - 2)]n n ( n - l ) !whic h is the same as (24) . Th e lef t -h and s ide of (36) i s the pro bab i l i ty t hat a tl eas t one co inc idence occurs in a p e rm uta t ion of 1 ,2 , . . . , n chosen a t r ando m.


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236 L. TAKACST h i s e v e n t c a n o c c u r in t w o m u t u a l l y e x c l u si v e w a y s : T h e f ir st n u m b e r c h o s e n i s1 w h i c h h a s p r o b a b i l i t y 1/n, o r t h e f i r s t n u m b e r i s d i f f e r e n t f r o m 1 , w h i c h h a sp r o b a b i l i t y (n - 1)In, a n d a t le a st o n e c o i n c i d e n c e o c c u r s a m o n g t h e r e m a i n i n g n- 1 e le m e n ts . T h e p r o o f o f (36 ) is c o m p l e t e i f w e p r o v e t h a t t h e n u m b e r o fp e r m u t a t i o n s o f 1, 2 , . . . , n i n w h i c h t h e f i rs t e l e m e n t i s a f i x e d in t e g e r i :# 1 a n d a tl ea s t o n e c o i n c id e n c e o c c u r s a m o n g t h e re m a i n i n g n - 1 e l e m e n ts is D ( n - 1 )- ( n - 2 ) ! + D ( n - 2 ) . T h e r e a r e t w o p o s s i b i l i t i e s : T h e ith e l e m e n t o f a p e r -m u t a t i o n is 1 o r :# 1. I n t h e f ir st ca s e, t h e n u m b e r o f p e r m u t a t i o n s i n w h i c h a tl ea s t o n e c o i n c id e n c e o c c u r s is D ( n - 2 ) . F o r i n e a c h p e r m u t a t i o n t h e fi rs te l e m e n t is i + 1 a n d t h e ith e l e m e n t is 1 a n d t h e r e m a i n i n g ( n - 2 ) e l e m e n t s c a n b ea r r a n g e d i n D ( n - 2 ) w a y s s u c h t h a t a t le a st o n e c o i n c id e n c e o c c u rs . In t h es e c o n d c a s e , t h e n u m b e r o f p e r m u t a t i o n s i n w h i c h a t l e a st o n e c o i n c i d e n c eo c c u r s is D ( n - 1 ) - ( n - 2 ) ! . F o r i n e a c h p e r m u t a t i o n t h e f i rs t e l e m e n t is i:t= 1 a n dt h e i h e l e m e n t is, sa y , j : # 1 , a n d t h e r e m a i n i n g ( n - 2 ) e l e m e n t s c a n b e a r r a n g e d i nD ( n - 1 ) - ( n - 2 ) ! w a y s s u c h t h a t a t le a s t o n e c o i n c i d e n c e o c c u r s a m o n g t h e m .T o s e e t h is l e t u s i n t e r c h a n g e t h e f i rs t a n d t h e i h e l e m e n t i n e a c h p e r m u t a t i o na n d r e m o v e t h e ith e l e m e n t w h i c h is i. S i n c e j :# 1 , t h e n e w f i rs t e l e m e n t c a n n o t b e1. T h u s i n ea c h p e r m u t a t i o n w e h av e n - 1 e le m e n ts a n d f ro m t h e D ( n - 1 )p e r m u t a t i o n s i n w h i ch a t l e as t o n e c o i n c i d e n c e o c c u rs w e s h o u l d r e m o v e ( n - 2 ) !p e r m u t a t i o n s i n w h i c h t h e f ir s t e l e m e n t is 1 . T h i s c o m p l e t e s t h e p r o o f o f (3 6)a n d (2 4). F i n a l l y , N . B E R N O U LL I i n d i c a t e d h o w (2 6) c a n b e d e d u c e d f r o m ( 2 4).

N IK O L A U S B E R N O UL L I's m a s t e r l y u s e o f c o n d i t i o n a l p r o b a b i l i t i e s i n t h ep r o o f o f (3 6) i s t r u l y r e m a r k a b l e .

T h e g e n e r a l s o l u t i o n o f t h e g a m e o f th i r te e n , P ( 1 3 , 4 , 1), c a n b e o b t a i n e d a s as t r a i g h t f o r w a r d e x t e n s i o n o f N . B E R N O U LL I's f o r m u l a (3 4). I t w o u l d b e i n t e r e s t -i n g to k n o w w h e t h e r M O N T M OR T o b t a i n e d ( 31 ) b e f o r e o r a f te r r e a d i n gBERNOULLI ' s r e m a r k s . I n a n y c a s e , i n a l e t t e r t o M O N T M O R T , d a t e d F e b r u a r y2 6 , 1 71 1, N . B E R N O U LL I [ 3 9 , p p . 3 0 8 - 3 1 4 ] g a v e a l s o t h e s o l u t i o n o f t h e g e n e r a lp r o b l e m ( p p . 3 0 8 - 3 0 9 ) . U n f o r t u n a t e l y , B E R N O U L L I ' s f o r m u l a c o n t a i n e d a n o v e r -s i g h t w h i c h M O N T M O R T [ 3 9 , p p . 3 1 5 - 3 2 3 ] e s s e n t i a l ly c o r r e c t e d in h is a n s w e r t oN. BERNOULLI , da t ed A pr i l 10 , 1711 . (See [ 39 , p . 315 ] . ) Af t e r w a r ds , BERNOULLI1-39, p p . 3 2 3 - 3 3 7 ] i n h is l e t t e r to M O N T M O R T , d a t e d N o v e m b e r 1 0, 1 71 1, g a v et h e c o r r e c t f o r m u l a ,(37) P(n,s, 1)= ~ (- 1 ) ~-1 (~ ) s~~= 1 n s( n s- 1 ) . . . ( n s - r + 1 ) 'f o r t h e p r o b a b i l i t y o f h a v i n g a t l e a st o n e c o i n c i d e n c e i n t h e g e n e r a l c a s e ([ 3 9,p p . 3 2 3 - 3 2 4 ] ) . I f n = 1 3 a n d s = 4 i n (3 7), w e g e t M O N T M O R T 'S r e s u l t (3 1).F o r m u l a ( 3 7) is a p a r t i c u l a r c a s e o f (2 0).

8 . T h e M e t h o d o f In c l u s io n a n d E x c l u s i o n

T h e p r o o f o f M O N T M O R T'S f o r m u l a (2 6), w h i c h N . B E R N O U LL I [ 3 9, p . 3 0 1 ]s e n t t o M O N T M O R T o n M a r c h 1 7, 17 1 0 i s v e r y s i g n i f ic a n t i n t h e t h e o r y o fp r o b a b i l i t y .


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Th e Pro blem of Coincidences 237By N. BERNOULLI's resul ts we can con clud e th at i f n events A~ ,A 2, . . . ,An

sa t i s fy the symmet ry proper t i e s(38) P{A h Ai2... Air} = P {AIA 2 . .. At}fo r l


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238 L. TAKACSf o r k = 1, 2 . . . . . n . D E M O IV R E'S p r o o f i s b a s e d o n t h e f o l lo w i n g f o r m u l a :(4 5) " { A 1 . .. A k d k + l . . . A , } = ~ ( - 1 ) r - k C - - ~ ) P { A I A 2 . . . A r }

r ~ k

w h i c h c a n b e p r o v e d i n t h e s a m e w a y a s ( 4 2 ) . A c t u a l l y , ( 4 5 ) i s a n i m m e d i a t ec o n s e q u e n c e o f (4 2). S i n c e o b v i o u s l y(46) P { ~ n = k } = ( ~ ) P { A 1 . . . A k . , ~ k + l . . . d n }f o r k = 1 , 2 . . . . . n , w e g e t ( 44) b y s u m m i ng ( 46 ) f o r k , k + 1 , . . . , n .

T h e a b o v e m e t h o d o f f i n d in g t h e d i s t r i b u t i o n o f ~ , w o r k s a l so i n t h e c as ew h e r e ( 38 ) d o e s n o t h o l d . I n t h i s c a s e t h e r i g h t - h a n d s i d e o f (4 4) is g i v e n b y ( 10 ).

F o r m u l a s (9 ) a n d (1 0) w e r e f o u n d i n 1 8 67 b y C . JO R D A N [-2 4]. I t w a s i n d e e da g r e a t a c h i e v e m e n t b y N . BE R N O U L LI a n d D E M O I V RE t o p r o v e (9 ) a n d (1 0) ina n i m p o r t a n t p a r t i c u l a r c a s e in 1 7 10 a n d 1 71 8 r e s p e ct iv e l y .

9 . E u l e r ' s C o n t r i b u t i o n

L . E U L E R w r o t e t w o p a p e r s o n t h e p r o b l e m o f c o i n c id e n c e s . I n h is p a p e r o f1 75 1 [ 1 3 ] h e g a v e a p r o o f fo r M O N T M O R T ' s f o r m u l a (2 6) b y u s i n g t h e s a m ea p p r o a c h a s N . B E R N O U LL I u s e d i n 1 7 1 0. E U LE R o b t a i n e d a l s o t h e l i m i t ( 27 ). I nh is p a p e r o f 1 7 79 L . E U L E R [ 1 4 ] p r o v e d t h a t i f Q ( n ) d e n o t e s t h e n u m b e r o fp e r m u t a t i o n s o f 1, 2 , . . . , n in w h i c h n o c o i n c i d e n c e o c c u r s , t h e n(47) Q (n) = (n - 1) [Q (n - 1) + (2 (n - 2)3f o r n > 2 w h e r e Q ( 0 ) = 1 a n d Q ( 1 ) = 0 . F r o m (47 ) h e d e ri v e d t h e e q u a t i o n

( - 1 )~(48) Q ( n ) - Q ( n - 1) -/7

f o r n = 1, 2 , . . . . S i n c e (4 7) is e q u i v a l e n t t o M O N T M O R T 'S e q u a t i o n (2 4) a n d s i n c e( 48 ) i m p l i e s ( 26 ) , EULER'S r e s u l t s p r ove MONTMORT'S f o r m ul a s ( 24 ) and ( 26 ) .

E U L ER 'S p r o o f o f (4 7) i s a l o n g t h e s a m e l in e s a s N . B E R N O U L L I'S p r o o f o f (2 4)i n 1 71 0. I n t h e s e t o f p e r m u t a t i o n s o f 1 , 2 , . . . , n i n w h i c h n o c o i n c i d e n c e o c c u r st h e f i r st e l e m e n t c a n o n l y b e i - - 2 , 3 . . . , n . T h e s e t i n w h i c h t h e f i r st e l e m e n t i si ( i + 1 ) c a n b e d i v i d e d i n t o t w o d i s j o i n t s u b s e ts . I n t h e f i r st s u b s e t t h e i t h e l e m e n ti s 1 , a n d i n t h e s e c o n d s u b s e t , t h e ith e l e m e n t i s j w h e r e j~ = 1. T h e n u m b e r o fp e r m u t a t i o n s i n th e f ir st s u b s e t is o b v i o u s l y Q ( n - 2 ) . F o r i f w e r e m o v e t h e f ir ste l e m e n t (/) a n d t h e i t l a e l e m e n t ( 1 ) f r o m e a c h p e r m u t a t i o n , t h e r e m a i n i n g n - 2e l e m e n ts c a n b e a rr a n g e d i n Q ( n - 2 ) w a y s s uc h t h a t n o c o i n c i d e n c e o c c ur s. T h en u m b e r o f p e r m u t a t i o n s i n th e s e c o n d s u b s e t is Q ( n - 1 ) . T o s ee th is l et usi n t e r c h a n g e t h e f i rs t e l e m e n t ( i) a n d t h e i h e l e m e n t ( j) i n e a c h p e r m u t a t i o n , a n dr e m o v e t h e n e w i th e l e m e n t (i) . T h e n u m b e r o f p e r m u t a t i o n s o b t a i n e d in t h is w a yis Q ( n - 1 ) . S i n c e i m a y t a k e t h e n - 1 v a l u e s 2 , 3 . . . . . n , w e g e t (4 7). S e e al soE. NETTO 1-42, pp . 66 -67 ] .


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The Problem of Coincidences 239In prov ing (24) N. BERNOULLI [39, pp. 301-302] essentia l ly demo nst ra te d

tha t(49) D(n) =(n - 1) [D(n - 1) + D ( n - 2)]f o r n > 2 whe r e D( 0) = 0 a nd D ( 1) = 1 . Equa t ions ( 47) a nd ( 49 ) a re e qu iva len t ;howev er , (47) can be pr ove d in a shorter way th an (49) .

10. Catalan's Contribution

MONTMORT'S fo rm ul a (3) was also pr ov ed by J .H . LAMBERT [33] in 1771.In 1812 P.S. LA PLA CE [34, pp. 217-225] , [35, pp. 219-228] der ivedN. BERNOULLI'S form ula (37) an d gave a n app rox ima ting expression for (37) . In1837 E. CATALAN [6] studied M od el I . He learn ed o f the results of EULER [13]and LAPLACE [34, pp. 217-225] only af ter his paper was f inished. He sho wed th atin formula (3) , Q(n) sat isf ies equ atio n (8) , and from (8) he der ived f orm ula (7)thu s p rov ing MONTMORT's resu lt (26) again. CATALAN'S pr oo f is on ly a slightva r i a n t o f N . BERNOULLI 's p r o o f or EULER'S.

CATALAN considered a lso a genera l iza t ion of Mod e l I by assuming tha tins tead of n we take out m cards f rom the box w here l_


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240 L. TAKACSw h e r e P ~ is t h e s e t o f a ll t h e n ! p e r m u t a t i o n s o f 1 , 2 . . . . n a n d 1(i~, i z . . . . , i , ) i s then u m b e r o f i n v e r s io n s i n t h e p e r m u t a t i o n ( ia , i2 . . . . , i, ). B y f o l lo w i n g T H . M U I R[ 4 0 , p . 1 9 ] w e d e f in e t h e p e r m a n e n t o f A , b y(54) P e r ( A , ) = ~ al l a212 . . . a , i .

( i 1 , i 2 , . . . , i n ) ~ P n

I n 1872 J . J . WEY RA U CH [53] s h o w e d t h a t i n ( 5 3 ) a n d i n ( 5 4 ) t h e r e a r e( ~ ) Q ( n - k ) t e r m s w h i c h c o n t a i n e x a c t ly k d i a g o n a l e l e m e n ts ( k = 0 , 1 . . . . , n )

w h e r e Q ( n ) is th e n u m b e r o f p e r m u t a t i o n s o f 1 , 2 , . . . , n in w h i c h n o c o i n c i d e n c eo c c u r s . Q ( n ) i s g iv e n b y ( 7) f o r n > 0 . S e e a l s o R . B A L T ZE R [ 1 ] . W E Y R A U C H 'sr e s u l t c a n a l s o b e f o r m u l a t e d i n t h e f o l l o w i n g w a y : I f A , ( x ) is an n x n m a t r i xw h o s e d i a g o n a l e l e m e n t s a r e x a n d w h o s e o f f - d i a g o n a l e l e m e n t s a r e 1 , t h e n(55)I n p a r t i c u l a r , w e h a v e(56)

k = O

P er (A , (0)) = Q (n).I n 1 89 1 G . D E L O N G C H A M P S [ 3 6 ] p r o v e d t h a t i f A , is a n n x n m a t r i x i n w h i c h

m d i a g o n a l e l e m e n t s a r e 0 a n d e v e r y o t h e r e l e m e n t i s d i f fe r e n t f r o m z e r o , t h e n( 53 ) and ( 54 ) con t a i n p r ec i s e l y Q ( n , m ) ( n - m ) ! n o n - z e r o t e r m s w h e r e Q ( n , m ) isg i ven by ( 52 ) .

D E L O NG C H A M PS [ 3 6 ] p r o p o s e d t h e p r o b l e m o f f i n d in g a f o r m u l a f o r t h en u m b e r o f n o n z e r o t e r m s i n (5 3) o r (5 4) f o r a m a t r i x A , i n w h i c h s o m e s e l e c te de l e m e n t s a r e e q u a l t o 0 . T h i s p r o b l e m w a s s o l v e d i n 1 8 9 1 b y C . A . L A I S A N T [ 3 2 ] ;s e e a ls o A . H O L T Z E [ 2 2 ] , E . N E T T O [ 4 2 , p p . 7 1 - 7 4 ] , [ 4 3 , p p . 7 1 - 7 4 ] , E . G . O LIO S[ 44 ] , J . RI O RD A N [ 46 , p. 184 ] , an d J . RY SER [ 47 , pp . 2 2 - 2 8 ] .

F o r m u l a s (5 5) a n d (5 6) a r e t r i v i a l ly t r u e . I f w e u s e M O N T M O R T 'S f o r m u l a ( 7)f o r Q ( n ) ( n > 0 ) , t h e n (5 5) a n d ( 56 ) c a n b e e x p r e s s e d i n e x p l i c i t f o r m s . C o n v e r s e l y ,w e c a n u s e ( 5 5 ) o r ( 5 6 ) f o r t h e d e t e r m i n a t i o n o f Q ( n) f o r n_> 0. I n 195 lJ . W . B O W E R [ 4 ] , a n d i n 1 97 3 J . J. JO H N S O N [ 2 3 ] p r o v e d M O N T M O R T 'S f o r m u l a(7 ) i n s u c h a w a y . I n 1 9 4 6 I a l so d e t e r m i n e d Q ( n ) b y u s i n g ( 5 6 ) . I a t t e n d e d t h ec la s se s o f P r o f e s s o r K . JO R D A N o n p r o b a b i l i t y t h e o r y w h e r e I l e a r n e d a b o u tM O N T M O R T ' s r e s u l t s (2 4) a n d (2 6) w h i c h w e r e s t a t e d w i t h o u t p r o o f i nM O N T M O R T ' s b o o k [ 3 8 ] . I f Q ( n ) i s d e f i n e d b y ( 2 ), t h e n b y M O N T M O R T ' s re s u l t sw e h a v e(57) Q (n) = (n - 1) [Q (n - 1) + Q (n - 2)3f o r n > 2 w h e r e Q ( 0 ) = 1 a n d Q ( 1 ) = 0, a n d c o n v e r s e l y (5 7) i m p l i e s (2 4) a n d (2 6). I to c c u r r e d t o m e i m m e d i a t e l y t h a t Q ( n ) c a n b e i n t e r p r e t e d a s th e n u m b e r o f t e r m sw h i c h c o n t a i n n o d i a g o n a l e l e m e n t s in a d e t e r m i n a n t o f o r d e r n , t h a t is, Q ( n ) ist h e p e r m a n e n t o f a n n n m a t r i x w h o s e d i a g o n a l e l e m e n t s a r e 0 a n d w h o s e o f f-d i a g o n a l e l e m e n t s a r e 1. B y u s in g t h i s i n t e r p r e t a t i o n o f Q ( n ) I g a v e t h e f o l l o w i n gp r o o f f o r ( 57 ): D e n o t e b y Q * ( n ) t h e p e r m a n e n t o f a n n x n m a t r i x w h o s e


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The Problem of Coincidences 241elements are 1 except the 2 nd, 3 ra, . .. ,n th diagona l e lements which a re 0 . Byexpanding both permanents according to the f i r s t co lumn we obta in(58) Q(n)= ( n - 1 )Q*(n- 1)a n d(59) O*(n) =Q (n - 1)+( n - 1) Q* (n- 1)= O(n - 1) + Q(n)for n> l . I f we express Q *( n- 1) in (58) by (59), we ge t (57) which was to beproved. I com mu nica ted m y resul t s to Professor JORDAN. In h is le tte r, da te dApril 28, 1946, Professor JORDAN wro te me tha t he in tend ed to refer to theseresul t s in h is for thcoming book on probabi l i ty theory which he had recent lyfinished; see K.JORDAN [25, p. 437], [26, p. 435].

1 2 . S o m e G e n e r a l i z a t i o n s

I f we assume tha t in M ode l I I the n um ber of l ' s , 2 's, . . . ,n ' s a re no t ne -cessar i ly the same, then we arr ive a t a gene ral izat ion o f the pro blem s discussed inthis paper . Such general m odels ha ve been studied by E .G. OLDS [44] ,J .A. GREENW OOD [19] , I . KAPLA NSKY [28] , [29] , I .L. BATT IN [3] ,I. KAPLANSKY & J. RIORDAN [30], and o thers .

There i s a la rge num ber of papers o n the s ta t i s tica l appl icat ions ofcoinc idence problems. H ere I men t ion only a paper by TH. YOUNG [55] w hichwas publ ished as ea r ly as 181 9. See an appra isa l of th is paper byM .G . KENDALL [31] .

The problem of coinc idences, i ts genera l iza tions a nd i ts va r ious appl ica t ionsare d iscussed in severa l books and in many papers . I ment ion the books ofW .A. WHITWORTH [54, pp, 100-117], E. LUCAS [37, pp. 211-215, 490-4 91],E. NET TO [42, pp. 66-7 4], [43, pp. 66-74 ], M. FRI;CHET [17, pp. 135-148],J. RIORDAN [46, pp. 57-62], H.J. RYSER [47, pp. 22-28 ] an d K . JORDAN [25, pp.431-4 41], [26, pp. 429-439]. In 194 3 M. FRI~CHET [18] pu blis hed a d etail edstu dy on coincid ences. See also M. CANT OR [5], A. CAYLEY [7], S. KANTOR[27], P. SEELHOFF [48], G. MUSS O [41], E. BATICLE [2], E. ESCARDO [12],M. FRgCHET [15], J. HAAG [20], I .B. HAAZ [21], M. ORTS [45], J. TOUCHARD[51] , [52] and others.

R e f e r e n c e s

1. R. BALTZER: Theorie und Anwendung der Determinanten. Third edition. Hirzel, Leip-zig, 1870.2. E. BATICLE:Le probl6me des rencontres. Comptes Rendus Acad. Sci. Paris 209 (1936)724-726 and 1891-1892.3. I.L. BATTIN: On the problem of multiple matching. Annals of M athematical Statistics13 (1942) 294-305.


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242 L. TAK~.CS4 . J .W . B O W E R: A n a p p l i c a t i o n o f d e t e r m i n a n t s t o t h e p r o b a b i l i ty o f m a t e d p a i rs . A m e r i -

c a n M a t h e m a t i c a l M o n t h l y 5 8 ( 1 9 5 1 ) 2 3 8 - 2 4 4 .5 . M . C A N T O R : U e b e r N o r m a l s t e l l e n . Z e i t s c h r i f t f /i T M a t h e m a t i k u n d P h y s i k 2 (1 85 7)

4 1 0 4 1 2 .6 . E . C A T A L A N: S o l u t i o n d ' u n p r o b l 6 m e d e p r o b a b i l i t 6 , r e l a t i f a u j e u d e r e n c o n t r e . J o u r n a ld e M a t h 6 m a t i q u e s P u r e s e t A p p l i q u 6 e s 2 ( 1 8 3 7 ) 4 6 % 4 8 2 .

7 . A . C A Y LE Y : N o t e o n M r . M u i r ' s s o l u ti o n o f a " p r o b l e m o f a r r a n g e m e n t " . P r o c e e d in g so f t h e R o y a l S o c i e t y o f E d i n b u r g h 9 (1 87 8) 3 8 8 - 3 9 l . [ T h e C o l l e c t e d M a t h e m a t i c a lP a p e r s o f A r t h u r C a y l e y . V o l . X . C a m b r i d g e U n i v e r s i t y P r e s s , 1 8 9 6 , p p . 24 9 2 51 .]

8. F . N . D A V ID : G a m e s , G o d s a n d G a m b l i n g . G r i ff in , L o n d o n , 1 96 2.9 . A . D E M O IV R E: T h e D o c t r i n e o f C h a n c e s : o r , A M e t h o d o f C a l c u l a ti n g t h e P r o b a b i l i t y

o f E v e n t s in P l a y . L o n d o n , 1 71 8.1 0. A . D E M O IV R E : T h e D o c t r i n e o f C h a n c e s : o r , A M e t h o d o f C a l c u l a t i n g t h e P r o b a b i l i t ie s

o f E v e n t s in P l a y . S e c o n d e d i t i o n . L o n d o n , 1 7 3 8 . [ R e p r i n t e d b y F r a n k C a s s a n d C o . ,L o n d o n , 1 9 6 7 . ]1 1. A . D E M O I V RE : T h e D o c t r i n e o f C h a n c e s : o r , A M e t h o d o f C a l c u l a t i n g t h e P r o b a b i l i t i e so f E v e n t s i n P l a y . T h i r d e d i ti o n . L o n d o n , 1 7 5 6 . [ R e p r i n t e d b y C h e l s e a , N e w Y o r k ,1967.]1 2. E . ES CA R D 0 : E1 p r o b l e m a d e l a s c o i n c i d e n c ia s . R e v i s t a M a t e m a t i c a H i s p a n o - A m e r i c a n a(4) 1 (1941) 20~214.1 3. L . E V L E R : C a l c u l d e l a p r o b a b i l i t 6 d a n s l e j e u d e r e n c o n t r e . M 6 m o i r e s d e l ' A c a d 6 m i ed e s S c i e n c e s d e B e r l i n , a n n 6 e 1 75 1, 7 ( 17 53 ) 2 5 5 -2 7 0 . [ L e o n h a r d i E u l e r i O p e r a O m n i a .SeT. I . Vo l . 7 . B .G. Te ubn er , Le ipz ig , 1923 , pp . 11 -25 . ]1 4. L . E tJ LE R : S o l u t i o q u a e s t i o n i s c u r i o s a e e x d o c t r i n a c o m b i n a t i o n u m . ( O c t o b e r 1 8,1 77 9) M 6 m o i r e s d e l ' A c a d 6 m i e d e s S c i e n c e s d e S t . - P 6 t e r s b o u r g ( 1 8 0% 1 8 1 0 ) 3 ( 18 11 )5 7 - 6 4 . [ L e o n h a r d i E u l e r i O p e r a O m n i a . S eT . I . V o l . 7 . B . G . T e u b n e r , L e i p z i g , 1 92 3,pp . 435-440 . ]1 5. M . E R I~ C H ET : A n o t e o n t h e " p r o b l ~ m e d e s r e n c o n t r e s " . A m e r i c a n M a t h e m a t i c a lMonth ly 46 (1939) 501 .

1 6. M . F R t~ C H ET : L e s p r o b a b i l i t 6 s a s s o c i6 e s / t u n s y s t 6 m e d ' 6 v 6 n e m e n t s c o m p a t i b l e s e td 6 p e n d a n t s . I e P aT t ie . t~ v 6 n e m e n t s e n h o m b r e f i n i fix e. A c t n a l i t 6 s S c i e n t i f iq u e s e t I n d u -s t r i e l l e s No . 859. H er m an n e t C ie , Pa r i s , 1940 .1 7. M . E R t~ C H ET : L e s p r o b a b i l i t 6 s a s s o c i 6 e s fi u n s y s t 6 m e d ' 6 v 6 n e m e n t s c o m p a t i b l e s e td 6 p e n d a n t s . I Ie P a T ti e. C a s p a r t i c u l i e r s e t a p p l i c a t i o n s . A c t u a l i t 6 s S c i e n t i f iq u e s e t In d u -s t r i e l l e s . No . 942 . H er m an n e t C i , Pa r i s , 1943 .1 8. M . F RI~ CH E T L e s s y s t b m e s d '6 v 6 n e m e n t s e t l e j e u d e s r e n c o n t r e s . R e v i s t a M a t e m a t i c aH i s p a n o - A m e r i c a n a ( 4 ) 4 (1 94 3) 9 5 - 1 26 .

1 9 . J . A . G R EE N W O OD : V a r i a n c e o f a g e n e r a l m a t c h i n g p r o b l e m . A n n a l s o f M a t h e m a t i c a lStat i s t ics 9 (1938) 56-59.

2 0. J . H A A G : S u r u n p r o b l ~ m e g 6 n 6 r a l d e p r o b a b i l i t 6 s e t s es d i v e r s e s a p p l i c a t i o n s .P r o c e e d i n g s o f t h e I n t e r n a t i o n a l C o n g r e s s o f M a t h e m a t i c i a n s , T o r o n t o , 1 92 4. T o r o n -to , 1928, pp. 659 674.2 1. I . B . H A A Z : A t a lf i lk o z f i s v a l d s z i n t i s 6 g 6 n e k M o n t m o r t - f 6 l e d i f f e r e n c i a e g y e n l e t e .B i z t o s i t / t s t u d o m f i n y i S z e m l e , B u d a p e s t , 1 94 2.2 2 . A . H O L T Z E : E i n i g e A u f g a b e n a u s d e r C o m b i n a t o r i k . A r c h i v d e r M a t h e m a t i k u n dPhysik (2) 11 (1892) 284-338.2 3. J .J . JO H N SO N : T h e p e r m a n e n t f u n c t io n a n d t h e p r o b l e m o f M o n t m o r t . M a t h e m a t i c sM aga z in e 46 (1973) 80 -83 .2 4 . C . J O R D A N : D e q u e l q u e s f o r m u l e s d e p r o b a b i l i t 6 . C o m p t e s R e n d u s A c a d . S c i . P a r i s65 (1867) 993-994.2 5. K . J O R D AN : F e j e z e t e k a K l a s s z i k u s V a l d s zi n ti s 6 g sz / tm i t /t s b 6 1 . ( H u n g a r i a n ) A k a d 6 m i a iK i a d o , B u d a p e s t , 1 9 5 6 .


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T h e P r o b l e m o f C o i n c i d e n c e s 2432 6 . K . J O R D A N : C h a p t e r s o n t h e C l a s s i c a l C a l c u l u s o f P r o b a b i l i t y . ( E n g l i s h t r a n s l a t i o n

o f t h e H u n g a r i a n o r i g i n a l p u b l i s h e d i n 1 9 56 .) A k a d 6 m i a i K i a d 6 , B u d a p e s t , 1 9 7 2.2 7. S . K A N TO R : P e r m u t a t i o n e n m i t b e s c h r~ i n k te r S t e l le n b e s e t z u n g . Z e i t s c h r if t ft i r M a t h e m a -

t ik und Phys ik 28 (1883) 379-383 .2 8. I . K A PL A NS K Y O n a g e n e r a l i z a t i o n o f t h e " p r o b l 6 m e d e s r e n c o n t r e s ". A m e r i c a n M a t h -e m a t i c a l M o n t h l y 4 6 ( 1 9 3 9 ) 1 5 % 1 6 1 .29 . I . K A P LA N S K Y : S y m b o l i c s o l u t i o n o f c e r t a in p r o b l e m s i n p e r m u t a t i o n s . B u l l e t i n o ft h e A m e r i c a n M a t h e m a t i c a l S o c i e t y $ 0 (1 94 4) 9 0 ( ~9 1 4 .

30. I . KAPLANSKY & J . R IO R D AN : M u l t i p l e m a t c h i n g a n d r u n s b y t h e s y m b o l i c m e t h o d .A n n a l s o f M a t h e m a t i c a l S t a t is t i c s 16 ( 19 45 ) 2 7 2 - 2 7 7 .3 1. M . G . K E N D A LL : S t u d i e s in t h e h i s t o r y o f p r o b a b i l i t y a n d s t a ti s ti c s . X V I I I . T h o m a sY o u n g o n c o i n c i d e n c es . B i o m e t r i k a $ $ (1 96 8) 2 4 9 - 2 5 0 .3 2. C . A . L A I SA N T : S u r l e s p e r m u t a t i o n s l i m i t 6 es . C o m p t e s R e n d u s A c a d . S c i . P a r i s 1 12(1891) 1047-1049.

3 3. J .H .L A M B E R T : E x a m e n d ' u n e e s p 6 c e d e s u p e r s t i t i o n r a m e n 6 e a u c a l c u l d e s p r o b a b i l i t 6 s .N o u v e a u x M 6 m o i r e s d e l ' A c a d 6 m i e R o y a l e d e s S c i e nc e s et B e l le s - L e t tr e s d e B e r l in ,ann6e 1771 (1773) 411-420.

34 . P~ S . LAPLACE: Th 6or i e A na ly t i qu e des P ro bab i l i t 6 s . Pa r i s , 1812 . [ -Rep r in t ed by Cu l tu r ee t C iv i l i sa t ion , Bruxe l l es , 1967 .]35 . P .S . LAPLACE: Th6 or i e An a ly t iqu e des P robab i l i t 6 s . Th i rd ed i t i on . Par i s , 1820 . [Rep r in -t e d : O e u v r e s C o m p l b t e s d e L a p l a c e . V o l . V I I . G a u t h i e r - V i l l a r s , P a r i s , 1 8 86 .]3 6. G . d e L O N G C H A M P S: S u r l e s d & e r m i n a n t s t r o u 6 s . J o u r n a l d e M a t h 6 m a t i q u e s s p ~ c ia l es

l ' u s a g e d e s c a n d i d a t s a u x l ~c ol es P o l y t e c h n i q u e s , N o r m a l e e t C e n t r a l e ( P a r i s ) (3 )$ (1891) %12, 29-32, 54-56, 85 87.

3 7. E . L U C A S : T M o r i e d e s N o m b r e s . I . G a u t h i e r - V i l l a r s , P a r i s , 1 8 91 .3 8. P . R . M O N T M O R T : E s s a y d ' A n a l y s e s u r le s J e u x d e H a z a r d . P a r i s , 1 7 08 .3 9. P . R . M O N T M O R T : E s s a y d ' A n a l y s e s u r le s J e u x d e H a z a r d . S e c o n d e d i t i o n , P a r i s ,

1713.4 0. T H . M U I R : A T r e a t i s e o n t h e T h e o r y o f D e t e r m i n a n t s . ( R e v i s e d a n d e n l a r g e d b yW . H , M E TZ LE R ) L o n g m a n s , G r e e n a n d C o ., 1 9 3 3. [ - R e p ri n te d b y D o v e r , N e w Y o r k ,1960.]

4 1. G . M u s s o : S u l l e p e r m u t a z i o n i r e l a ti v e a d u n a d a t a . R i v i s t a d i M a t e m a t i c a 4 ( 18 94 )109-119.

4 2. E . N ~T T O: L e h r b u c h d e r C o m b i n a t o r i k . T e u b n e r , L e i p z i g , 1 90 1.4 3. E . N ET T O : L e h r b u c h d e r C o m b i n a t o r i k . S e c o n d e d i t i o n . T e u b n e r , L e i p z i g , 1 9 2 7.

[ R e p r i n t e d b y C h e l s e a , N e w Y o r k , 1 9 6 4 . ]4 4. E . G . O L D S: A m o m e n t - g e n e r a t i n g f u n c t i o n w h i c h i s u s e f u l i n s o l v i n g c e r t a i n m a t c h i n gp r o b l e m s . B u l l e t i n o f th e A m e r i c a n M a t h e m a t i c a l S o c i e t y 4 4 (1 93 8) 4 0 7 - 4 1 3 .4 5. M . O R T S : S o b r e u n a g e n e r a l i z a c i 6 n d e l p r o b l e m a d e l a s c o i n c i d e n c i a s . R e v i s t a M a t e m a -t i c a H i s p a n o - A m e r i c a n a 12 (1 93 7) 1 7 -1 9 .4 6. J. R IO R D A N : A n I n t r o d u c t i o n t o C o m b i n a t o r i a l A n a l y s i s . J o h n W i l e y a n d S o n s , N e w

York , 1958 .4 7. H . J . R Y S ER : C o m b i n a t o r i a l M a t h e m a t i c s . T h e C a r u s M a t h e m a t i c a l M o n o g r a p h s .N o . 1 4 . T h e M a t h e m a t i c a l A s s o c i a t i o n o f A m e r i c a , 1 9 6 3 .4 8. P . S E EL H O FF : U e b e r a l l g e m e i n e u n d a b s o l u t e P e r m u t a t i o n e n . A r c h i v d e r M a t h e m a t i kund Physik (2) 1 (1884) 97-101.4 9. L . TA K ~.C S O n t h e m e t h o d o f i n c lu s i o n a n d e x c l u s i o n . J o u r n a l o f th e A m e r i c a n S t a t i s t i-ca l Ass oc ia t ion 62 (1967) 102-113 .5 0. I . T O D H U N TE R : A H i s t o r y o f t h e M a t h e m a t i c a l T h e o r y o f P r o b a b i l i t y f r o m t h e T i m eo f P a s c a l to t h a t o f L a p l a c e. C a m b r i d g e , 1 8 6 5 . [ R e p r i n te d b y C h e l s e a , N e w Y o r k ,1949.]5 1. J . T O U C H A R D : R e m a r q u e s s u r l e s p r o b a b i l i t 6 s to t a l e s e t s u r l e p r o b l 6 m e d e s r e n c o n t r e s .A nn ale s de la Soci6t6 Scie nt i f iqu e de Brux el les . S6r . A $3 (1933) 12(~134.


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244 L. TAKKCS5 2. J .T O U C H A R D: S u r u n p r o b lG m e d e p e r m u t a t i o n s . C o m p t e s R e n d u s A c a d . S c i . P a r is

198 (1934) 631-633.5 3. J . J . W E Y R AU C H : Z u r T h e o r i e d e r D e t e r m i n a n t e n . J o u r n a l f t i r d i e r e in e u n d a n g e w a n d t e

M a t h e m a t i k 7 4 ( 1 8 7 2 ) 2 7 3 - 2 7 6 .5 4 . W . A . W H I T W O R T H : C h o i c e a n d C h a n c e w i t h O n e T h o u s a n d E x e r c i s e s . F i f t h e d i t i o n .D e i g h t o n B e ll , C a m b r i d g e , 1 9 0 1 . [ R e p r i n te d b y H a f n e r P u b l i s h i n g C o . , N e w Y o r k ,1959.]

5 5. T H . Y O U N G : R e m a r k s o n t h e p r o b a b i l i t i e s o f e r r o r i n p h y s i c a l o b s e r v a t i o n s , a n d o nt h e d e n s i t y o f t h e e a r t h , c o n s i d e r e d , e s p e c i a l l y w i t h r e g a r d t o t h e r e d u c t i o n o f e x p e r i -m e n t s o n t h e p e n d u l u m . ( I n a l e t t e r t o H . K A T ER ) P h i l o s o p h i c a l T r a n s a c t i o n s o f t h eR o y a l S o c i e t y o f L o n d o n 1 09 (1 81 9) 7 0 - 9 5 .

D e p a r t m e n t o f M a t h e m a t i c s & S t a ti s ti c sC a s e W e s t e r n R e s e r v e U n i v e r s i t y

C l e v e l a n d , O h i o(Received Septem ber 14, 1979)

n julio 3_takacs - montmort problem

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