Question

Each of $3$ committees has $1$ vacancy which is to be filled from a group of $6$ people. Find the number of ways the $3$ vacancies can be filled if Each person can serve on atmost $1$ committee.

Answer 1

Select one committee out of $3{=}^3{C}_1$

Select one person out of $6{=}^6{C}_1$

Select one committee out of $2{=}^2{C}_1$

Select one person out of $5{=}^5{C}_1$

Select last committee$=1$

Select one person out of $4{=}^4{C}_1$

Number of ways the $3$ vacancies can be filled if Each person can serve on atmost $1$ committee${=}^6{C}_1{\times }^2{C}_1{\times }^5{C}_1\times 1{\times }^4{C}_1$

$=\frac{6!}{5!1!}\times \frac{2}{1!}\times \frac{5!}{4!1!}\times 1\times \frac{4!}{3!1!}$

$=6\times 2\times 5\times 4=240$ ways.