counting subsets of a set: combinations lecture 33 section 6.4 tue, mar 27, 2007

Counting Subsets of a Set: Combinations

Lecture 33

Section 6.4

Tue, Mar 27, 2007

Lotto South

In Lotto South, a player chooses 6 numbers from 1 to 49.

Then the state chooses at random 6 numbers from 1 to 49.

The player wins according to how many of his numbers match the ones the state chooses.

See the Lotto South web page.

Lotto South

There are C(49, 6) = 13,983,816 possible choices.

Match all 6 numbersThere is only 1 winning combination.Probability of winning is

1/13983816 = 0.00000007151.

Lotto South

Match 5 of 6 numbersThere are 6 winning numbers and 43 losing

numbers.Player chooses 5 winning numbers and 1

losing numbers.Number of ways is C(6, 5) C(43, 1) = 258.Probability is 0.00001845.

Lotto South

Match 4 of 6 numbersPlayer chooses 4 winning numbers and 2

losing numbers.Number of ways is C(6, 4) C(43, 2) =

13545.Probability is 0.0009686.

Lotto South

Match 3 of 6 numbersPlayer chooses 3 winning numbers and 3

losing numbers.Number of ways is C(6, 3) C(43, 3) =

246820.Probability is 0.01765.

Lotto South

Match 2 of 6 numbersPlayer chooses 2 winning numbers and 4

losing numbers.Number of ways is C(6, 2) C(43, 4) =

1851150.Probability is 0.1324.

Lotto South

Match 1 of 6 numbersPlayer chooses 1 winning numbers and 5

losing numbers.Number of ways is C(6, 1) C(43, 5) =

3011652.Probability is 0.4130.

Lotto South

Match 0 of 6 numbersPlayer chooses 6 losing numbers.Number of ways is C(43, 6) = 2760681.Probability is 0.4360.

Lotto South

Note also that the sum of these integers is 13983816.

Note also that the lottery pays out a prize only if the player matches 3 or more numbers.Match 3 – win $5.Match 4 – win $75.Match 5 – win $1000.Match 6 – win millions.

Lotto South

Given that a lottery player wins a prize, what is the probability that he won the $5 prize?

P(he won $5, given that he won)

= P(match 3)/P(match 3, 4, 5, or 6)

= 0.01765/0.01864

= 0.9469.

Example

Theorem (The Vandermonde convolution): For all integers n 0 and for all integers r with 0 r n,

Proof: See p. 362, Sec. 6.6, Ex. 18.

r

k r

n

kr

rn

k

r

0

Another Lottery

In the previous lottery, the probability of winning a cash prize is 0.018637545.

Suppose that the prize for matching 2 numbers is… another lottery ticket!

Then what is the probability of winning a cash prize?

Lotto South

What is the average prize value of a ticket? Multiply each prize value by its probability

and then add up the products:$10,000,000 0.00000007151 = 0.7151$1000 0.00001845 = 0.0185$75 0.0009686 = 0.0726$5 0.01765 = 0.0883$0 0.9814 = 0.0000

Lotto South

The total is $0.8945, or 89.45 cents (assuming that the big prize is ten million dollars).

A ticket costs $1.00. How large must the grand prize be to make

the average value of a ticket more than $1.00?

Another Lottery

What is the average prize value if matching 2 numbers wins another lottery ticket?

Permutations of Sets with Repeated Elements

Theorem: Suppose a set contains n1 indistinguishable elements of one type, n2 indistinguishable elements of another type, and so on, through k types, where

n1 + n2 + … + nk = n.

Then the number of (distinguishable) permutations of the n elements is

n!/(n1!n2!…nk!).

Proof of Theorem

Proof: Rather than consider permutations per se,

consider the choices of where to put the different types of element.

There are C(n, n1) choices of where to place the elements of the first type.

Proof of Theorem

Proof: Then there are C(n – n1, n2) choices of

where to place the elements of the second type.

Then there are C(n – n1 – n2, n3) choices of where to place the elements of the third type.

And so on.

Proof, continued

Therefore, the total number of choices, and hence permutations, is

C(n, n1) C(n – n1, n2) C(n – n1 – n2, n3) … C(n – n1 – n2 – … – nk – 1, nk)

= …(some algebra)…

= n!/(n1!n2!…nk!).

Example

How many different numbers can be formed by permuting the digits of the number 444556?

60!1!2!3

!6

Example

How many permutations are there of the letters in the word MISSISSIPPI?

How many for VIRGINIA? How many for INDIVISIBILITY?

34650!1!2!4!4

!11

Poker Hands

Two of a kind. Two pairs. Three of a kind. Straight. Flush. Full house. Four of a kind. Straight flush. Royal flush.