Elements of Combinatorics

1

Permutations

• (Weak Definition) A permutation is usually understood to be a sequence containing each element from a finite set once, and only once.

• The concept of sequence is distinct from that of a set, in that the elements of a sequence appear in some order: the sequence has a first element (unless it is empty), a second element (unless its length is less than 2), and so on. In contrast, the elements in a set have no order; {1, 2, 3} and {3, 2, 1} are different ways to denote the same set.

2

Permutations

• (General Definition) A permutation is an ordered sequence of elements selected from a given finite set, without repetitions, and not necessarily using all elements of the given set.

• For example, given the set of letters {C, E, G, I, N, R}, some permutations are ICE, RING, RICE, NICER, REIGN and CRINGE

3

Permutations• The total number of different permutations of n elements of a set with the cardinality n is

4

( ) !P n n

( 1) ( 2) ... 2 1( ) !n n nP n n

Permutations

• The number of different (ordered) permutations (arrangements) of r elements selected from n is

5

( 1) ... ( 1)!

( , )( )!

rn

nP n n n rn r A

n r

!( , )

( )!rn

nP n r A

n r

Combinatorics: where we need it?

• For example, if students have today Calculus (C), Physics (P), and Discrete Mathematics (D) classes. How we can calculate the probability that D is the first class?

• The following 6 arrangements are possible: CPD, CDP, PCD, PDC, DCP, DPC. Two of them are desirable: DCP and DPC. Thus, if all events are equiprobable, then the probability is 2/6=1/3.

6

The number of subsets of a set

• Theorem. If n is any nonnegative integer, then a set of the cardinality n (a set with n elements) has exactly 2n subsets.

7

Combinations

• A combination is an un-ordered collection of distinct elements, usually of a prescribed size and taken from a given set.

• Given a set S, a combination of elements of S is just a subset of S, where, as always for (sub)sets the order of the elements is not taken into account (two lists with the same elements in different orders are considered to be the same combination). Also, as always for (sub)sets, no elements can be repeated more than once in a combination; this is often referred to as a "collection without repetition"

8

Combinations

• The number of different (not ordered) combinations of r elements selected from n is the number of all possible permutations (arrangements) of r objects selected from n divided by the number of all possible permutations of r objects r!:

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!,

!( )!!rn

rnA

r

nC n r C

r n r

, rnP n r A

The number of subsets of a set

• Theorem. Let S be a set containing n elements, where n is a nonnegative integer. If r is an integer such that , then the number of subsets of S containing exactly r elements is

10

0 nr

( , ) !( , )

! ! !( )!

rr nn

AP n r nC n r C

r r r n r

Combinatorics: where we need it?• Let students take during this semester

Calculus (C), Physics (P), and Discrete Mathematics(D) classes, two classes/day. How we can calculate the probability that D and P are taken at the same day?

• There are 3 different combinations of 2 objects selected from 3: (CP=PC), (CD=DC), (DP=PD). One of them is desirable: (DP=PD). Thus, the probability is 1/3.

11

1st Property of Combinations

• Theorem. If r and n are integers such that , then

12

1 r n

11 1

, 1, 1 1,r r rn n n

C n r C n r C n r

C C C

2nd Property of Combinations

• Theorem. If r and n are integers such that , then

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1 r n

, ,r n rn n

C n r C n n r

C C

Binomial Theorem

• Binomial Theorem (I. Newton). Let x and y be the variables, and n is a nonnegative integer. Then are the coefficients of the binomial decomposition (binomial coefficients):

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, 0,...,rn rC n

0 1 1 2 2 2 ...

... ...

n n n nn n n

r n r r n nn n

x y C x C x y C x y

C x y C y

Pascal’s Triangle

15

00

0 11 1

0 1 22 2 2

0 1 2 33 3 3 3

1

1 1

1 2 1

1 3 3 1

C

C C

C C C

C C C C

Pascal’s Triangle

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Pascal’s Triangle

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000 11 10 1 22 2 20 1 2 33 3 3 30 1 2 3 44 4 4 4 4

C

C C

C C C

C C C C

C C C C C

Combinatorics: where we need it?

• Tournament problem. Suppose there are n chess players and they participate in the tournament, where everybody has to play with all other participants exactly 1 time. How many parties will be played in this tournament?

• How many parties will be played if each participant has to play with others twice?

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2nC

2nA