In how many ways ,can 20 identical apples be divided among 5 persons?

Options

A. 12626
B. 10626
C. 9286
D. 11626

Answer

To solve the problem of how many ways 20 identical apples can be divided among 5 persons, we can use the "stars and bars" theorem from combinatorics. Here's a step-by-step solution:

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( n \) be the number of identical apples, which is 20.
- Let \( r \) be the number of persons, which is 5.

2. **Apply the Stars and Bars Theorem**:
- According to the stars and bars theorem, the number of ways to distribute \( n \) identical items (apples) among \( r \) distinct groups (persons) is given by the formula:
\[ \binom{n + r - 1}{r - 1} \]
- In our case, we need to calculate:
\[ \binom{20 + 5 - 1}{5 - 1} = \binom{24}{4} \]

3. **Calculate \( \binom{24}{4} \)**:
- The binomial coefficient \( \binom{24}{4} \) can be calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
- Therefore:
\[ \binom{24}{4} = \frac{24!}{4!(24 - 4)!} = \frac{24!}{4! \cdot 20!} \]

4. **Simplify the Calculation**:
- We can simplify \( \frac{24!}{20!} \) as follows:
\[ \frac{24 \times 23 \times 22 \times 21 \times 20!}{20!} = 24 \times 23 \times 22 \times 21 \]
- Now, we need to divide this by \( 4! \):
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
- Thus:
\[ \binom{24}{4} = \frac{24 \times 23 \times 22 \times 21}{24} \]

5. **Final Calculation**:
- Cancel the 24 in the numerator and denominator:
\[ 23 \times 22 \times 21 \]
- Now calculate:
\[ 23 \times 22 = 506 \]
\[ 506 \times 21 = 10626 \]

6. **Conclusion**:
- Therefore, the number of ways to divide 20 identical apples among 5 persons is **10,626**.