Question

In a town council there are $25$ councillors and $10$ aldermen; how many committees can be formed each consisting of $5$ councillors and $3$ aldermen?

Answer 1

Selection of $5$ councillors from $25$ is given by ${}^{25}{C}_5=\frac{25!}{5!20!}=\frac{25\times 24\times 23\times 22\times 21}{120}=53,130$

Selection of $3$ aldermen from $10$ is given by ${}^{10}{C}_3=\frac{10!}{3!7!}=\frac{10\times 9\times 8}{6}=120$

Total number of ways are $53,130\times 120=63,75,600$