Question

The number of ways of distributing $8$ identical balls in $3$ distinct boxes so that none of the boxes is empty are?

• A. $3^8$
• B. $21$
• C. $5$
• D. ${}^8{C}_3$

Answer 1

The correct option is B

$21$

Explanation for the correct option(s)

Find the number of ways.

Consider the $8$ identical bolls are distributed in $3$ distinct boxes when none of the boxes are empty

Then the remaining balls are $5$ and the remaining boxes are $3$.

We know that the number of ways can be expressed without any condition that is

$$\begin{gathered} {}^{n+r-1}{C}_{r-1}={}^{5+3-1}{C}_{3-1} \\ {}^{n+r-1}{C}_{r-1}={}^7{C}_2 \end{gathered}$$

We know that

${}^n{C}_r=\frac{n!}{r!\left(n-r\right)!}$

Then,

$$\begin{gathered} {}^7{C}_2=\frac{7!}{2!\left(7-2\right)!} \\ {}^7{C}_2=\frac{7\times 6\times 5!}{2\times 1\times 5!} \\ {}^7{C}_2=7\times 3 \\ {}^7{C}_2=21 \end{gathered}$$

Hence, Option B is correct.