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 in the committee. In how many ways can it be done?

Answer 1

Now in question it is asked to find the number of ways a committee can be formed so that there are, at least 3 men mean that there can be 3 men or 4 men or all of the 5 committee members as men. So, we will have three cases.
Case 1: When there are 3 men, then there will be 2 women.
Case 2: When there are 4 men, then there will be 1 woman.
Case 3: When there are 5 men, then there will be 0 women.
And for solving these cases we will use The formula for selecting r different things from n different things called as combination and is given as${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$ .
The formula for finding the number of ways for arranging n different things is given as$=n!$, where $n!=n(n-1)(n-2)....3.2.1$.

Complete step-by-step answer:
As mentioned in the question, there will be 3 cases for calculating the number of ways to form a committee is as follows

Case 1: 3 Men, 2 Women
For selecting 3 men out of 7 and 2 women out of 6 to form thee 5 people committee, we can use the formulas given in the hint as follows
$={}^{7}C{}_{3}\times {}^{6}C{}_{2}$
$=35\times 15$
$=525$
Case 2: 4 Men, 1 Woman
For selecting 4 men out of 7 and 1 woman out of 6 to form thee 5 people committee, we can use the formulas given in the hint as follows
$={}^{7}C{}_{4}\times {}^{6}C{}_{1}$
$=35\times 6$
$=210$
Case 3: 5 Men, 0 Women
For selecting 5 men out of 7 and 0 women out of 6 to form thee 5 people committee, we can use the formulas given in the hint as follows
$={}^{7}C{}_{5}\times {}^{6}C{}_{0}$
$=21\times 1$
$=21$

Total =525+210+21 =756

Hence, the answer total number of ways to form a 5 member committee is 756.

Note: The students can make an error if they don’t know the fundamental theorem and the formula for selecting r different things from n different things which is given as ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$.
Try not to make any calculation errors as this will change the final answer.