The number of ways in which $21$ identical apples can be distributed among three children such that each child gets at least $2$ apples, is

Let us initially distribute $2$ apples to each children.

So, the remaining $15$ apples are to be distributed such that each children can get any number of apples.

So, the required number of ways will be given by $N={}^{n+r-1}C_{r-1}$.

$\Rightarrow N={}^{15+3-1}C_2$

$\Rightarrow N={}^{17}C_2$

$\Rightarrow N=\frac{17\times 16}{2}$

$\Rightarrow N=136$