Question

We have $19$ identical gems available with us which are needed to be distributed among $A,B$ and $C$ such that $A$ always gets an even number of gems. The number of ways this can be done is

Answer 1

Answer: 110

$A+B+C=19$
Case I: If $A\doteq 0$ then number of ways $={}^{20}{C}_1$
Case II : If $A=2$ then number of ways $={}^{18}{C}_1$
Case III : If $A=3$ then number of ways $={}^{16}{C}_1$
$\begin{array}{cccc}:& :& :& :\\ :& :& :& :\end{array}$
Case $X$ : If $A=18$ then number of ways $={}^2{C}_1$
$\therefore$ total no. of ways $={}^{20}{C}_1+{}^{18}{C}_1+{}^{16}{C}_1+\dots +{}^2{C}_1=110$