Question

In how many ways a committee of $6$ people can be chosen from a panel of $13$ people, if a certain person from $13$ people must be on the committee?

• A. ${}^{13}{C}_5$
• B. ${}^{12}{C}_5$
• C. ${}^{13}{C}_7$
• D. ${}^{12}{C}_7$

Answer 1

Answer: (B), (D)

${}^{12}{C}_7$

Total number of people $n=13$

We need to form a committee of $6$ people from a panel of $13$ people.

But, it is given that a certain person is already chosen from $13$ persons and is also a member of the committee.

This implies that we need to choose only $5$ people from a panel of $12$ since a person is already selected from $13$ people to form the committee.

$\therefore r=5$

$5$ people can be chosen from $12$ people in ${}^{12}{C}_5$ ways. If we do ${}^{13}{C}_5$ then the person whose selection is certain will be repeated. Hence, we cannot do it.

But,

${}^{12}{C}_5={}^{12}{C}_{12-5}$ ($\because {}^n{C}_r={}^n{C}_{n-r}$)

${}^{12}{C}_5={}^{12}{C}_7=\frac{12!}{5!7!}$


The number of ways in which a committee of $6$ people can be chosen from a panel of $13$ people if a certain person from $13$ people must be on the committee is ${}^{12}{C}_5$ or ${}^{12}{C}_7$.

Therefore, option (b.) and (d.) are the correct answers.