The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange…

Given,

We have to distribute $20$ identical oranges to $3$ children,

Let children $A$ get $O_A$ oranges, children $B$ gets $O_B$ oranges and children $C$ gets $O_C$ oranges,

So, $O_A+O_B+O_C=20$

Now this can be distributed using the multinomial theorem,

So, Number of ways will be,

$=$coefficient of ${\mathrm{x}}^{20}$ in ${\left(\mathrm{x}+{\mathrm{x}}^2+\dots .+x^{18}\right)}^3$

$=$coefficient of ${\mathrm{x}}^{20}$ in $x^3{\left(1+x+{\mathrm{x}}^2+\dots .x^{17}\right)}^3$

$=$coefficient of $x^{17}$ in ${\left(\frac{1-x^{18}}{1-x}\right)}^3$

$=$coefficient of ${\mathrm{x}}^{17}$ in $(1-\mathrm{x}{)}^{-3}$

$={}^{19}\mathrm{C}_2=171$

As we know that expansion of $(1-\mathrm{x}{)}^{-3}$ is $1+{}^{3}C_{1}x+{}^{4}C_2{x}^2+{}^{5}C_3{x}^3+......+{}^{19}C_{17}{x}^{17}........\infty$

Note: This is bonus question as here $20$ oranges are considered identical but in question it is considered distinct.