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Chapter 1: Q. 1.12 (page 20)

A committee of 6 people is to be chosen from a group consisting of 7 men and 8 women. If the committee must consist of at least 3 women and at least 2 men, how many different committees are possible?

Short Answer

Expert verified

The possible no. of committees are3430.

Step by step solution

01

Step 1. Given information.

Total no. of men =7

Total no. of women =8

Total no. of members in committee =6

Minimum no. of women in committee =3

Minimum no. of men in committee =2

There can be two ways of choosing the committee, either 3men and 3women or 2 men and 4 women.

02

Step 2. Find the no. of ways of choosing 3 men and 3 women for committee.

3men out of 7can be chosen in role="math" localid="1649155856082" 73ways=7!3!4!=35

3women out of 8can be chosen in 83ways=8!3!5!=56

Therefore, the no. of ways of choosing a committee of 3 men and 3 women=35脳56=1960

03

Step 3. Find the no. of ways of choosing 2 men and 4 women for committee.

2out of 7men can be chosen in 72ways=7!2!5!=21

4out of8women can be chosen in84ways=8!4!4!=70

Therefore, the no. of ways of choosing a committee of 2 men and 4 women=21脳70=1470

04

Step 4. Find the possible no. of committees.

Therefore, the possible no. of committees are=1960+1470=3430.

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Most popular questions from this chapter

Consider the following combinatorial identity:

鈭�k=1nknk=n路2n-1

(a) Present a combinatorial argument for this identity by considering a set of npeople and determining, in two ways,

the number of possible selections of a committee of any size and a chairperson for the committee.

Hint:

(i) How many possible selections are there of a committee of size kand its chairperson?

(ii) How many possible selections are there of a chairperson and the other committee members?

(b) Verify the following identity for n=1,2,3,4,5:

localid="1648098528048" 鈭�k=1nnkk2=2n-2n(n+1)

For a combinatorial proof of the preceding, consider a set of n people and argue that both sides of the identity represent

the number of different selections of a committee, its chairperson, and its secretary (possibly the same as the chairperson).

Hint:

(i) How many different selections result in the committee containing exactly kpeople?

(ii) How many different selections are there in which the chairperson and the secretary are the same?

(answer: n2n鈭�1.)

(iii) How many different selections result in the chairperson and the secretary being different?

(c) Now argue that

localid="1647960575612" 鈭�k=1nnkk3=2n-3n2(n+3)

Expandx1+2x2+3x34.

If 12people are to be divided into 3committees of respective sizes 3,4,and 5,how many divisions are possible?

From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. How many

different committees are possible if

(a) 2 of the men refuse to serve together?

(b) 2 of the women refuse to serve together?

(c) 1 man and 1 woman refuse to serve together?

A total of nstudents are enrolled in a review course for the actuarial examination in probability. The posted

results of the examination will list the names of those who passed, in decreasing order of their scores. For instance, the posted result will be 鈥淏rown, Cho鈥� if Brown and Cho are the only ones to pass, with Brown receiving the higher score. Assuming that all scores are distinct (no ties), how many posted results are possible?
See all solutions

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