In how many ways can 12 things be equally divided among 4 persons ?

Answer

To solve the problem of how many ways 12 things can be equally divided among 4 persons, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We need to divide 12 things equally among 4 persons. Each person will receive an equal number of items.

2. **Calculate Items per Person**:
Since there are 12 items and 4 persons, we divide 12 by 4:
\[ \text{Items per person} = \frac{12}{4} = 3 \]
Each person will receive 3 items.

3. **Selecting Items for Each Person**:
We will use combinations to determine how many ways we can select items for each person.

- **First Person**: The first person can choose 3 items from the 12 available items. The number of ways to do this is given by:
\[ \binom{12}{3} \]

- **Second Person**: After the first person has chosen their 3 items, there are 9 items left. The second person can choose 3 items from these 9:
\[ \binom{9}{3} \]

- **Third Person**: After the second person has chosen their items, there are 6 items left. The third person can choose 3 items from these 6:
\[ \binom{6}{3} \]

- **Fourth Person**: Finally, the fourth person will choose the remaining 3 items from the last 3 items:
\[ \binom{3}{3} \]

4. **Total Ways to Distribute**:
The total number of ways to distribute the items is the product of the combinations calculated above:
\[ \text{Total Ways} = \binom{12}{3} \times \binom{9}{3} \times \binom{6}{3} \times \binom{3}{3} \]

5. **Calculate Each Combination**:
Using the formula for combinations:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
We can compute each term:
- \(\binom{12}{3} = \frac{12!}{3! \cdot 9!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220\)
- \(\binom{9}{3} = \frac{9!}{3! \cdot 6!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84\)
- \(\binom{6}{3} = \frac{6!}{3! \cdot 3!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20\)
- \(\binom{3}{3} = 1\)

6. **Final Calculation**:
Now, we multiply these values together:
\[ \text{Total Ways} = 220 \times 84 \times 20 \times 1 \]
- Calculate \(220 \times 84 = 18480\)
- Then, \(18480 \times 20 = 369600\)

Thus, the total number of ways to equally divide 12 things among 4 persons is **369600**.