Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • In a group of 10 people, consisting of 5 men and 5 women, a committee of 4 people needs to be formed. However, the committee must have at least 2 men and at least 2 women. How many different ways c

In a group of 10 people, consisting of 5 men and 5 women, a committee of 4 people needs to be formed. However, the committee must have at least 2 men and at least 2 women. How many different ways can the committee be formed?

 

Option: 1

400


Option: 2

600


Option: 3

800


Option: 4

200


Answers (1)

best_answer

To solve this problem, we can break it down into cases:

Case 1: 2 men and 2 women

We need to select 2 men from the 5 available, which can be done in C(5, 2) ways. Similarly, we need to select 2 women from the 5 available, which can also be done in C(5, 2) ways. Therefore, the number of ways to form the committee with 2 men and 2 women is \mathrm{C\left ( 5,2 \right )\times C\left ( 5,2 \right )}.

Case 2: 3 men and 1 woman

We need to select 3 men from the 5 available, which can be done in C(5, 3) ways. Similarly, we need to select 1 woman from the 5 available, which can be done in C(5, 1) ways. Therefore, the number of ways to form the committee with 3 men and 1 woman is \mathrm{C\left ( 5,3 \right )\times C\left ( 5,1 \right )}.

Case 3: 1 man and 3 women

We need to select 1 man from the 5 available, which can be done in C(5, 1) ways. Similarly, we need to select 3 women from the 5 available, which can be done in C(5, 3) ways. Therefore, the number of ways to form the committee with 1 man and 3 women is \mathrm{C\left ( 5,1 \right )\times C\left ( 5,3 \right )}.

To find the total number of ways to form the committee, we need to sum up the results from all three cases:

\mathrm{\text{Total number of ways}=C(5,2) \times C(5,2)+C(5,3) \times C(5,1)+C(5,1) \times C(5,3)}

 Evaluating each term:

\mathrm{ C(5,2)=5 ! /(2 ! \times(5-2) !)=10}
\mathrm{ C(5,3)=5 ! /(3 ! \times(5-3) !)=10}
\mathrm{ C(5,1)=5 ! /(1 ! \times(5-1) !)=5}

Plugging in the values:

\mathrm{ 10 \times 10+10 \times 5+5 \times 10}

= 100 + 50 + 50

= 200

Therefore, there are 200 different ways to form the committee with at least 2 men and at least 2 women from a group of 10 people consisting of 5 men and 5 women.

Evaluating this expression, we can find the total number of different ways the committee can be formed.

 

Posted by

shivangi.bhatnagar

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2026 Registration LIVE: NTA session 1 form link at jeemain.nta.nic.in; application deadline
anam-khan

JEE 2026: IIT admissions show major shift toward AI, data science, away from core engineering courses
anam-khan

JEE Main 2026: NIT Delhi cut-offs for BTech courses show ever-increasing demands for AI, computer skills

Latest Question