JEE Main 2023 — Mathematics Algebra

Given,

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

Let children $A$A get ${O}_{A}$OA​ oranges, children $B$B gets ${O}_{B}$OB​ oranges and children $C$C gets ${O}_{C}$OC​ oranges,

So, ${O}_{A}+{O}_{B}+{O}_{C}=20$OA​+OB​+OC​=20

Now this can be distributed using the multinomial theorem,

So, Number of ways will be,

$=$=coefficient of ${x}^{20}$x20 in ${(x+{x}^{2}+\ldots .+{x}^{18})}^{3}$(x+x2+….+x18)3

$=$=coefficient of ${x}^{20}$x20 in ${x}^{3}{(1+x+{x}^{2}+\ldots .{x}^{17})}^{3}$x3(1+x+x2+….x17)3

$=$=coefficient of ${x}^{17}$x17 in ${(\frac{1-{x}^{18}}{1-x})}^{3}$(1−x1−x18​)3

$=$=coefficient of ${x}^{17}$x17 in $(1-x{)}^{-3}$(1−x)−3

$$=C219=171$$=C219=171

As we know that expansion of $(1-x{)}^{-3}$(1−x)−3 is $1+C13x+C24{x}^{2}+C35{x}^{3}+......+C1719{x}^{17}........\infty$1+C13x+C24x2+C35x3+......+C1719x17........∞

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