12 different toys are to be distributed to three children equally. In how many ways this can be done?

Answer

To solve the problem of distributing 12 different toys equally among three children, we need to follow these steps:

### Step-by-Step Solution:

1. **Determine the Distribution**: Since we have 12 different toys and we want to distribute them equally among 3 children, each child will receive \( \frac{12}{3} = 4 \) toys.

2. **Use the Formula for Grouping**: The number of ways to distribute \( n \) different items into \( r \) groups of \( k \) items each (where \( n = r \times k \)) can be calculated using the formula:
\[ \frac{n!}{k!^r} \]
In our case, \( n = 12 \), \( k = 4 \), and \( r = 3 \). Thus, we can substitute these values into the formula.

3. **Apply the Formula**:
\[ \text{Number of ways} = \frac{12!}{4!^3} \]
Here, \( 4!^3 \) accounts for the fact that the toys in each group of 4 are indistinguishable among themselves.

4. **Account for Indistinguishable Groups**: Since the groups are indistinguishable (the order of children does not matter), we need to divide by the number of ways to arrange the 3 groups, which is \( 3! \). Therefore, the formula becomes:
\[ \text{Number of ways} = \frac{12!}{4!^3 \times 3!} \]

5. **Calculate Factorials**:
- Calculate \( 12! = 479001600 \)
- Calculate \( 4! = 24 \), so \( 4!^3 = 24^3 = 13824 \)
- Calculate \( 3! = 6 \)

6. **Substitute Values into the Formula**:
\[ \text{Number of ways} = \frac{479001600}{13824 \times 6} \]

7. **Perform the Multiplication**:
\[ 13824 \times 6 = 82944 \]

8. **Final Calculation**:
\[ \text{Number of ways} = \frac{479001600}{82944} = 5760 \]

### Final Answer:
The total number of ways to distribute the 12 different toys equally among the three children is **5760**.

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