Total number of ways in which 20 mangoes can be distributed among 4 persons is :

Options

A. `""^(20)C_3`
B. `""^(23)C_3`
C. `""^(24)C_4`
D. `(20!)/(4!)`

Answer

To solve the problem of distributing 20 mangoes among 4 persons, we can use the "stars and bars" theorem from combinatorics. This theorem helps us find the number of ways to distribute indistinguishable objects (mangoes) into distinguishable boxes (persons).

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( n \) be the total number of mangoes, which is 20.
- Let \( r \) be the number of persons, which is 4.

2. **Apply the Stars and Bars Theorem**:
- According to the stars and bars theorem, the number of ways to distribute \( n \) indistinguishable objects into \( r \) distinguishable boxes is given by the formula:
\[ \binom{n + r - 1}{r - 1} \]
- In our case, we need to substitute \( n = 20 \) and \( r = 4 \) into the formula.

3. **Calculate the Values**:
- Substitute the values into the formula:
\[ \binom{20 + 4 - 1}{4 - 1} = \binom{23}{3} \]

4. **Compute the Binomial Coefficient**:
- Now, we need to calculate \( \binom{23}{3} \):
\[ \binom{23}{3} = \frac{23!}{3!(23-3)!} = \frac{23!}{3! \cdot 20!} \]
- This simplifies to:
\[ \binom{23}{3} = \frac{23 \times 22 \times 21}{3 \times 2 \times 1} = \frac{10626}{6} = 1771 \]

5. **Final Answer**:
- Therefore, the total number of ways to distribute 20 mangoes among 4 persons is **1771**.