The number of ways in which 5 prizes can be distributed among 4 boys when every boy can take one or more prizes is

Options

A. 1024
B. 625
C. 120
D. 600

Answer

To solve the problem of distributing 5 prizes among 4 boys where each boy can receive one or more prizes, we can use the "stars and bars" theorem from combinatorics.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to distribute 5 identical prizes among 4 distinct boys. Each boy can receive any number of prizes, including none.

2. **Using the Stars and Bars Theorem**:
The stars and bars theorem states that if we have \( n \) identical items (stars) to distribute into \( k \) distinct groups (bars), the number of ways to do this is given by the formula:
\[ \text{Number of ways} = \binom{n + k - 1}{k - 1} \]
Here, \( n \) is the number of prizes (5), and \( k \) is the number of boys (4).

3. **Applying the Formula**:
In our case:
- \( n = 5 \) (the prizes)
- \( k = 4 \) (the boys)

Plugging these values into the formula:
\[ \text{Number of ways} = \binom{5 + 4 - 1}{4 - 1} = \binom{8}{3} \]

4. **Calculating the Binomial Coefficient**:
Now we need to calculate \( \binom{8}{3} \):
\[ \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!} \]
Simplifying this:
\[ = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

5. **Conclusion**:
Therefore, the number of ways in which 5 prizes can be distributed among 4 boys is **56**.