Question

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

• A. $15$
• B. $25$
• C. $35$
• D. $40$

Answer 1

The correct option is C

$35$

Explanation for correct option:

Step1. distributing things among the people:

Let $p,q,r\&s$are four different numbers of things that are distributed to each person.

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

If each $4$persons gets at least $3$ things, then total things distributed will be

$$\begin{gathered} =3\times 4 \\ =12 \end{gathered}$$

Step 2. Distributing remaining item among a group:

Now, there are $4$ things left.

We can distribute n identical objects among r groups in $C_{r-1}^{n+r-1}$ways.

Therefore, the number of ways, 4 identical objects can be distributed among 4 persons is

$$\begin{gathered} =C_{4-1}^{4+4-1} \\ =C_3^7 \\ =\frac{7!}{3!\times 4!} \\ =\frac{7\times 6\times 5\times 4!}{3\times 2\times 4!} \\ =35 \end{gathered}$$

Therefore, Option $\left(C\right)$ is the correct answer.