Question

A father has to distribute $12$ books to his $4$ children such that each child has to receive atleast one book. In how many ways can he do it?

• A. $165$
• B. $666$
• C. $408$
• D. $1024$

Answer 1

Answer: (A)

$165$

There are $12$ books to be distributed to $4$ children.

Let $x_1,x_2,x_3,x_4$ represent the number of books given to each child.

$\Rightarrow x_1+x_2+x_3+x_4=12$ $,$ $x_1,x_2,x_3,x_4\ge 1$

Let $x_i-1=y_i,$ $x_i-1\ge 0\Rightarrow y_i\ge 0$

The equations becomes,

$\Rightarrow y_1+y_2+y_3+y_4=12-4=8$

Total combinations is given by ${}^{n+r-1}{C}_{r-1}$

Here, $n=8$ $,$ $r=4$

$\Rightarrow {}^{8+4-1}{C}_{4-1}={}^{11}{C}_3$

$=165.$

Hence, the answer is $165.$