The number of ways of giving 20 distinct oranges to 3 children such...

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 ${x}^{20}$ in ${\left(x + {x}^{2} + \ldots . + {x}^{18}\right)}^{3}$

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

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

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

$= C2 19 = 171$

As we know that expansion of $( 1 - x{)}^{- 3}$ is $1 + C1 3 x + C2 4 {x}^{2} + C3 5 {x}^{3} + . . . . . . + C17 19 {x}^{17} . . . . . . . . \infty$

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