Find the number of ways in which $16$ identical things can be distributed among $4$ persons, if each person gets at least $3$ things.

The correct option is C

$35$

Explanation for the correct option:

Step:1 Find out the number of things left after giving each persons $3$ things.

Let $p,q,r,s$ are the number of things to be distributed among person.

Therefore, $p+q+r+s=16$.

If each person gets at least $3$ things, then $3\times 4=12$ things are already distributed.

Now, $16-12=4$things are left.

Step:2 Find out the ways the that are left can be distributed among $4$ persons.

We can distribute $n$number of things can be distributed among $r$ persons in ${\mathit{C}}_{\mathbf{r}\mathbf{-}\mathbf{1}}^{\mathbf{n}\mathbf{+}\mathbf{r}\mathbf{-}\mathbf{1}}$ways.

Therefore, $4$ things can be distributed among $4$ persons in ways:

$\Rightarrow C_{4-1}^{4+4-1}$

$$\begin{gathered} =C_3^7 \\ =\frac{7!}{3!\times 4!} \\ =\frac{7\times 6\times 5\times 4!}{4!\times 3\times 2} \\ =35 \end{gathered}$$

Therefore, Option(C) is the correct answer.