From a Group of 7 Men and 6 Women, Five Persons are to be Selected GMAT Problem Solving

Question: From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?

1. 564
2. 645
3. 735
4. 756
5. 566

Correct Answer: D

Solution and Explanation:
Approach Solution 1:
The problem statment states that:
Given:

• There are 7 men and 6 women. 
• Five persons are to be selected to form a committee so that at least 3 men are there on the committee.

​Find out :

• The number of ways it can be done.

This is a question from permutation and combination where we have to select five persons from a group of 7 men and 6 women.
In order to solve the problem, the candidate should remember the following formulas for permutation and combination.
Permutation of n objects taken r at a time:\(^nP_r\)= n! / (n-r)!
Combination of n objects taken r at a time: \(^nC_r\) = n! / ( (n-r)! * r!)
where x! = x*(x-1)*(x-2)*......(3).(2).(1)
It is necessary to have at least 3 men on the committee.
So we can choose 3,4 or 5 men from the group of 7 men and accordingly we’ll pick women.
The total sum of all the possibilities will be the answer.
We can choose 3 from 7 men and 2 from 6 women in \(^7C_3*{^6C_2}\) ways
We can choose 4 from 7 men and 1 from 6 women in \(^7C_4*{ ^6C_2}\) ways
We can choose 5 from 7 men and 0 from 6 women in \(^7C_5\)ways.

Therefore the total number of ways to select 5 people having at least 3 men is = \(^7C_3*{ ^6C_2}+{^7C_4}*{^6C_1}+{^7C_5}\)
= 35*15 + 35*6 + 21 = 756
Hence, the number of ways it can be done = 756 ways.

Approach Solution 2:
The problem statment implies that:
Given:

• There are 7 men and 6 women. 
• Five persons are to be selected to form a committee so that at least 3 men are there on the committee.

​Find out :

• The number of ways it can be done.

Further, the problem can be calculated in terms of the total number of ways in which no of men is less than 3. Then subtracting the latter from the total ways.

If we select all females then the total number of ways = \(^6C_5\)
If we select 4 women and 1 man, the total number of ways = \(^6C_4*{^7C_1}\)
If we select 3 women and 2 men, the total number of ways = \(^6C_3*{ ^7C_2}\)
Total number of ways when three men are not there =
\(^6C_5+({ ^6C_4*{^7C_1}})+({^6C_3}*{^7C_2})\) = 531
Total number of ways to select 5 people from a group of 13 people = \(^{13}{C_5}\) = 1287
Total number of ways of selecting 5 people so that there are at least 3 men is =
1287 - 531 = 756

Hence, the number of ways it can be done = 756 ways.

Approach Solution 3:
The problem statment informs that:
Given:

• There are 7 men and 6 women. 
• Five persons are to be selected to form a committee so that at least 3 men are there on the committee.

​Find out :

• The number of ways it can be done.

The problem statement suggests 3 scenarios in which at least 3 men can be selected for the 5-person committee.

The three ways could be as follows:
3 men and 2 women OR 4 men and 1 woman OR 5 men.
Let’s calculate the number of ways to select the committee in each scenario.

Scenario 1: 3 men and 2 women
The number of ways to select 3 men: 7C3 = (7 x 6 x 5)/3! = (7 x 6 x 5)/(3 x 2 x 1) = 35
The number of ways to select 2 women: 6C2 = (6 x 5)/2! = (6 x 5)/(2 x 1) = 15
Therefore, the number of ways to select 3 men and 2 women is 35 x 15 = 525.

Scenario 2: 4 men and 1 woman
The number of ways to select 4 men: 7C4 = (7 x 6 x 5 x 4)/4! = (7 x 6 x 5 x 4)/(4 x 3 x 2 x 1) = 35
The number of ways to select 1 woman: 6C1 = 6
Therefore, the number of ways to select 4 men and 1 woman is 35 x 6 = 210.

Scenario 3: 5 men
The number of ways to select 5 men: 7C5 = (7 x 6 x 5 x 4 x 3)/5!
= (7 x 6 x 5 x 4 x 3)/(5 x 4 x 3 x 2 x 1) = 42/2 = 21

Therefore, the number of ways to select a 5-person committee with at least 3 men is
= 525 + 210 + 21 = 756

Hence, the number of ways it can be done = 756 ways.

“From a group of 7 men and 6 women, five persons are to be selected”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Prep Plus”. The GMAT Problem Solving questions encourage the candidates to analyse data and solve numerical problems. GMAT Quant practice papers help the candidates to get familiar with several sorts of questions that will enable them to score better in the exam.

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