Question

In how many ways can $20$ oranges be given to four children if each child should get at least one orange?

• A. $869$
• B. $969$
• C. $973$
• D. None of these

Answer 1

The correct option is B $969$

The maximum number of oranges that a child can get $=20-3=17.$

Thus, the problem is equivalent to finding the number of integral solutions to the equation $x_1+x_2+x_3+x_4=20$

where $1\le x_1,x_2,x_3,x_4\le 17,$ and $x_1,x_2,x_3,x_4$ denote the number of oranges given to the four children.

Hence, the required number of ways is

$=$Coefficient of $x^{20}$ in ${(x+x^2+x^3+...+x^{17})}^4$

$=$coefficient of $x^{16}$ in ${(1+x+x^2+...+x^{16})}^4$

$=$coefficient of $x^{16}$ in ${(1-x^{17})}^4{(1-x)}^{-4}$

$=$coefficient of $x^{16}$ in ${(1-x)}^{-4}$

${=}^{16+4-1}{C}_{16}$

${=}^{19}{C}_3=969.$