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

Answer: (B)

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 ${\left(x+x^2+x^3+\dots +x^{17}\right)}^4$
$=$ coefficient of $x^{16}$ in ${\left(1+x+x^2+\dots +x^{16}\right)}^4$
$=$ coefficient of $x^{16}$ in ${\left(1-x^{17}\right)}^4(1-x{)}^{-4}$
$=$ coefficient of $x^{16}$ in $(1-x{)}^{-4}$
$={}^{16+4-1}{C}_{16}$
$={}^{19}{C}_3=969$