Question

At an election $3$ wards of a town are canvassed by $4,5$ and $8$ men respectively. If there are $20$ men volunteering the number of ways they can be allotted to the different wards is?

• A. $\frac{20!}{3!4!5!}$
• B. $\frac{12!}{3!4!5!}$
• C. $\frac{20!}{3!4!5!8!}$
• D. $\frac{12!}{3!4!5!8!}$

Answer 1

The correct option is C $\frac{20!}{3!4!5!8!}$

Number of ways of allotting $4$ mens to the first ward is ${}^{20}{C}_4$

Number of ways of allotting $5$ mens from the remaining $16$ mens to the second ward is ${}^{16}{C}_5$

Number of ways of allotting $8$ mens from the remaining $11$ mens to the third ward is ${}^{11}{C}_8$

The number of ways to split the group $={}^{20}{C}_4{}^{16}{C}_5{}^{11}{C}_8=\frac{20!}{4!16!}\times \frac{16!}{11!5!}\times \frac{11!}{8!3!}=\frac{20!}{3!4!5!8!}$

This is the required answer.