In how many different ways can 6 identical balls be distributed to 4 different boxes , when each box can have any number of balls ?

Options

A. 12
B. 18
C. 84
D. 88

Answer

To solve the problem of distributing 6 identical balls into 4 different boxes, we can use the "stars and bars" theorem, which is a common combinatorial method for distributing indistinguishable objects into distinguishable boxes.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to distribute 6 identical balls into 4 different boxes. Each box can hold any number of balls, including zero.

2. **Applying the Stars and Bars Theorem**: The stars and bars theorem states that if we have \( n \) indistinguishable objects (stars) to place into \( k \) distinguishable boxes (with bars separating them), the number of ways to do this is given by the formula:
\[ \text{Number of ways} = \binom{n + k - 1}{k - 1} \]
where \( n \) is the number of indistinguishable objects (balls) and \( k \) is the number of distinguishable boxes.

3. **Identifying Variables**: In our case:
- \( n = 6 \) (the number of balls)
- \( k = 4 \) (the number of boxes)

4. **Substituting into the Formula**:
\[ \text{Number of ways} = \binom{6 + 4 - 1}{4 - 1} = \binom{9}{3} \]

5. **Calculating the Binomial Coefficient**:
\[ \binom{9}{3} = \frac{9!}{3!(9 - 3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84 \]

6. **Conclusion**: Therefore, the number of different ways to distribute 6 identical balls into 4 different boxes is **84**.

### Final Answer:
The number of different ways to distribute 6 identical balls into 4 different boxes is **84**.

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