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

Hint: At least 3 men mean that there can be 3 men or 4 men or all of the 5 committee members as men. When there are 3 men, then there will be 2 women.Similarly for 4 and 5 men, then there will be 1 and 0 women.Use the formula for selecting r different things from n different things is given as \[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\].Select 3 men from group of 7 i.e .${}^{7}C{}_{3}$ and 2 women from group of 6 i.e.${}^{6}C{}_{2}$ to form committee of 5 people and multiply these combinations.Similarly calculate for other two cases and add all those values to get required answer.

Complete step-by-step answer:
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
\[\begin{align}
  & ={}^{7}C{}_{3}\times {}^{6}C{}_{2} \\
 & =35\times 15 \\
 & =525 \\
\end{align}\]

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
\[\begin{align}
  & ={}^{7}C{}_{4}\times {}^{6}C{}_{1} \\
 & =35\times 6 \\
 & =210 \\
\end{align}\]

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
\[\begin{align}
  & ={}^{7}C{}_{5}\times {}^{6}C{}_{0} \\
 & =21\times 1 \\
 & =21 \\
\end{align}\]

$Total =525+210+21 =756$

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
\[{}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}\].Here the word atleast means we have to take more than 3 men i.e upto 5 men we can take from a group of 7 as the committee is to be formed only of 5 people.