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

$A+B+C=19$
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 $={ }^{16} C_{1}$
$\begin{matrix}: & : & : & : \\ : & : & : & :\end{matrix}$
Case $X$ : If $A=18$ then number of ways $={ }^{2} C_{1}$
$\therefore $ total no. of ways $={ }^{20} C_{1}+{ }^{18} C_{1}+{ }^{16} C_{1}+\ldots+{ }^{2} C_{1}=110$