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Q.
In how many ways can $20$ oranges be given to four children if each child should get at least one orange?
Permutations and Combinations
Solution:
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 $1 \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 17$, 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 $\left(x+x^{2}+x^{3}+\ldots+x^{17}\right)^{4}$
$=$ coefficient of $x^{16}$ in $\left(1+x+x^{2}+\ldots+x^{16}\right)^{4}$
$=$ coefficient of $x^{16}$ in $\left(1-x^{17}\right)^{4}(1-x)^{-4}$
$=$ coefficient of $x^{16}$ in $(1-x)^{-4}$
$={ }^{16+4-1} C_{16}$
$={ }^{19} C_{3}=969$