The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes i

Q: Wait, why can't we just use $3^8$?

That is a common trap! The formula $3^8$ assumes that the balls are distinct (like labeled balls). Because our balls are identical, the order in which they are placed into the boxes does not create a new arrangement. We must use the Stars and Bars method instead.

Q: Why is it $n-1$ gaps and not $n$?

Great observation. If we have $n$ balls, there are $n+1$ possible positions for dividers (including the ends). However, the constraint 'no box is empty' forbids us from placing dividers at the very beginning or the very end. Thus, we are restricted to the $n-1$ internal gaps between the balls.

Q: What if empty boxes were allowed?

If empty boxes were allowed, we would use the formula $^{n+r-1}C_{r-1}$. In that scenario, we would have $^{8+3-1}C_{3-1} = ^{10}C_2 = 45$ ways. Always check the problem statement for the 'no empty box' constraint!

Question

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

Options:

(A) ^8C_3

(B) 21

(C) 3^8

(D) 5

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are diving into a classic problem that sits at the heart of combinatorics. Imagine you are in a lab, and you have $8$ identical blue balls.

You have $3$ distinct boxes—let's call them Box A, Box B, and Box C. Your mission is to distribute these balls such that no box is left empty. It sounds simple, but the beauty lies in the systematic approach.

### The Stars and Bars Intuition

To solve this, we use a powerful technique called the **'Stars and Bars'** method. Imagine the $8$ balls lined up in a row. To divide them into $3$ distinct groups, we need to place $2$ dividers.

Think of it like this: one cut creates two pieces, and two cuts create three pieces. However, there is a catch! The constraint is that no box can be empty.

If we place a divider at the very beginning or the very end of the line, one of the boxes would end up with zero balls. Therefore, we must place our $2$ dividers strictly in the internal gaps between the balls.

### Counting the Gaps

Let's visualize the gaps. With $8$ balls, there are exactly $8 - 1 = 7$ internal gaps. We need to choose $2$ of these $7$ gaps to place our dividers.

Because the balls are identical, the only thing that matters is which gaps we choose. This is a classic combination problem: we are choosing $2$ items from a set of $7$. Mathematically, this is expressed as $^7C_2$.

### The General Formula

We can generalize this for any $n$ identical items and $r$ distinct boxes. The number of ways to distribute them such that no box is empty is given by the formula:

$$^{n-1}C_{r-1}$$

Substituting our values, $n=8$ and $r=3$, we get:

$$^{8-1}C_{3-1} = ^{7}C_{2}$$

### The Final Calculation

Now, let's calculate the value of $^7C_2$. Recall the formula for combinations:

$$^nC_r = \frac{n!}{r!(n-r)!}$$

Applying this, we have:

$$^7C_2 = \frac{7!}{2!(7-2)!} = \frac{7!}{2! 	imes 5!}$$

We can simplify this by writing $7!$ as $7 	imes 6 	imes 5!$. The $5!$ terms cancel out, leaving us with:

$$\frac{7 	imes 6}{2 	imes 1} = \frac{42}{2} = 21$$

There are exactly **21 ways** to distribute the balls. This is the power of mathematical thinking—transforming a physical problem into a clean, elegant calculation. Keep practicing, and you will master these concepts in no time!

Animated Solution for Mathematics - Permutations and Combinations: The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

Select Answer:

Visualized Solution

Visualizing the Setup: $8$ Identical Balls

The Gap Method (Stars and Bars)

Counting the Available Gaps

Choosing Gaps for $2$ Dividers

Formulating with Combinations: $^{7} C_{2}$

The General Formula: $^{n - 1} C_{r - 1}$

Expanding $^{7} C_{2}$

Simplifying the Factorials

Final Calculation: $21$ Ways

Summary & Key Takeaway

The Sigma Insight: Combinations and Selection

Analyzing the Setup

The Stars and Bars Intuition

Counting the Gaps

The General Formula

The Final Calculation

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