Case1:- 7C3 × 6C2 = 525

case2:- 7C4 × 6C1 = 245

case3:- 7C5 × 6C0 = 21
total number of ways = 525 + 245 + 21

= 791

Problem:
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?

Solution:

To solve this problem, we can use the concept of combinations.

Step 1: Calculate the total number of ways to select 5 people from a group of 13:
We have a group of 13 people (7 men and 6 women), and we need to select 5 people from this group. The total number of ways to do this is given by the combination formula:

C(13, 5) = 13! / (5!(13-5)!) = 13! / (5!8!) = (13 * 12 * 11 * 10 * 9) / (5 * 4 * 3 * 2 * 1) = 1287

So, there are 1287 ways to select 5 people from the group of 13.

Step 2: Calculate the number of ways to select 5 people with at least 3 men:
To find the number of ways to select 5 people with at least 3 men, we can calculate the number of ways to select 5 people with exactly 3 men and the number of ways to select 5 people with exactly 4 men, and then add them together.

Case 1: Selecting 5 people with exactly 3 men:
We have 7 men to choose from, so we can select 3 men out of 7 in C(7, 3) = 7! / (3!(7-3)!) = 7! / (3!4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35 ways.

We have 6 women to choose from, so we can select 2 women out of 6 in C(6, 2) = 6! / (2!(6-2)!) = 6! / (2!4!) = (6 * 5) / (2 * 1) = 15 ways.

Therefore, there are 35 * 15 = 525 ways to select 5 people with exactly 3 men.

Case 2: Selecting 5 people with exactly 4 men:
We have 7 men to choose from, so we can select 4 men out of 7 in C(7, 4) = 7! / (4!(7-4)!) = 7! / (4!3!) = (7 * 6 * 5) / (4 * 3 * 2 * 1) = 35 ways.

We have 6 women to choose from, so we can select 1 woman out of 6 in C(6, 1) = 6! / (1!(6-1)!) = 6! / (1!5!) = 6 ways.

Therefore, there are 35 * 6 = 210 ways to select 5 people with exactly 4 men.

Step 3: Calculate the total number of ways to select 5 people with at least 3 men: