STAT 430/510 Lecture 2: Permutations and Combinations, Study notes of Statistics

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STAT 430/510 Probability

Hui Nie

Lecture 2

May 27th, 2009

Review

The basic principle of counting provides us a powerful tool for counting the number of ways that things can happen. An ordered sequence of k distinct objects taken from a set of n objects is called a permutation of size k. The total number of permutations of size k from n objects is given by Pk,n =

n! (n − k)!

Example

From a group of 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed? Solution:

What if 2 of the men are feuding and refused to serve on the committee together? Solution:

Example

Consider a set of n antennas of which m are defective and n − m are functional and assume that all of the defectives and all of the functionals are considered indistinguishable. How many linear orderings are there in which no two defectives are consecutive?

Solution:

n − m + 1 m

The Binomial Theorem

The Binomial Theorem

(x + y)n^ =

∑^ n

r = 0

n r

xr^ yn−r

n r

is called binomial coefficients.

Example

Find the coefficient of x^2 y in (x + y)^3

Solution:

Example

A set of n distinct items is to be divided into r distinct groups of respective sizes∑ n 1 , n 2 , · · · , nr , where r i= 1 ni^ =^ n. How many different divisions are possible? Solution:( n n 1

) ( (^) n − n 1 n 2

) ( (^) n − n 1 − n 2 n 3

( (^) n − n 1 − n 2 − · · · − nr − 1 nr

= (^) n 1 !nn 2 !!···nr!

Multinomial Coefficients

If

∑r i= 1 ni^ =^ n, define

n n 1 , n 2 , · · · , nr

by

( n n 1 , n 2 , · · · , nr

n! n 1 !n 2! · · · nr! ( n n 1 , n 2 , · · · , nr

are known as multinomial coeffcients. ( n n 1 , n 2 , · · · , nr

represents the number of possible

divisions of n distinct objects into r distinct groups of respective sizes n 1 , n 2 ,..., nr.

Example

In order to play a game of basket ball, ten children at a plyground divided themselves into two teams of 5 each. How many different games are possible?

Solution:

  10 5 , 5

 

2! =^

10! 5! 5! 2! =^126

The Multinomial Theorem

The Multinomial Theorem

(x 1 + x 2 + · · · + xr )n^ =

(n 1 ,··· ,nr ):n 1 +···+nr =n

( (^) n n 1 , n 2 , · · · , nr

x 1 n 1 x 2 n 2 · · · x rnr

Example

Find the coefficient of x^6 yz in ( 3 x^2 + y + z)^5

Solution:

33 = 5!^3

3 3! 1! 1!