Question

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

• A. ${}^{20}P_4\times {}^{20}P_5\times {}^{20}P_8$
• B. ${}^{20}C_4\times {}^{20}C_5\times {}^{20}C_8$
• C. ${}^{20}C_4\times {}^{16}C_5\times {}^{11}C_8$
• D. $\frac{1}{3!}{}^{20}C_4\times {}^{16}C_5\times {}^{11}C_8$

Answer 1

The correct option is C

${}^{20}C_4\times {}^{16}C_5\times {}^{11}C_8$

Explantion for the correct options:

Finding the number of ways men can be allotted to different ward is:

Number of ways allotting $4$men to the first ward is ${}^{20}C_4$

Number of ways of allotting $5$ men from the remaining $16$ men to the second ward is ${}^{16}C_5$

Number of ways of allotting 8 men from the remaining 11 men to the third ward is ${}^{11}C_8$

Therefore the number of ways to split the group =${}^{20}C_4\times {}^{16}C_5\times {}^{11}C_8$

Hence correct option is $\left(D\right)$