Committe. Suppose that a department contains 10 men and 15 women. How manyways are there to form a committee with six members if it must have the same number ofmen and women

There 54,600 ways to form a committee with six members with the same number of men and women.

Step-by-step explanation:

Since we are to form a committee of six members with the same number of men and women, that is

In the committee, we must have 3 men and 3 women.

First we will find the number of ways to form a committee of 3 men out of 10 men and then 3 women out of 15 women, we will then use the product rule to determine the number of ways to form the committee of 3 men and 3 women.

From the combination formula

Where n is the number of items is set

and r is the the number of items selected from set

To form a committee of 3 men out of 10 men

Then,

n = 10

and r = 3

∴  

Then,

To form a committee of 3 women out of 15 men

n = 15

r = 3

∴  

Applying the product rule: If there are n ways to do A and m ways to do B, then the number of ways to do A and B is n × m.

Therefore, the number of ways to form a six members committee of 3 men and 3 women is

120 × 455 = 54600

Hence, there 54,600 ways to form a committee with six members with the same number of men and women.

the first option is the answer.