In how many ways can 12 children be divided into three equal sized identical (indistinguishable) groups?

To solve this problem, we need to determine how many ways 12 children can be divided into three equal-sized groups, where the groups are indistinguishable (identical). This is a classic example of a partitioning problem, which can be addressed using a formula for dividing \(n\) objects into \(k\) indistinguishable groups of equal size.

Here, we have:
- 12 children
- 3 groups, each containing \( \frac{12}{3} = 4 \) children.

### Step-by-Step Solution:
1. **Divide 12 children into three groups**:
   If the groups were distinguishable, we would first compute the number of ways to divide the 12 children into three groups of size 4. The formula to divide \(n\) items into \(k\) groups of size \(a, b, c\) (such that \(a + b + c = n\)) is given by the multinomial coefficient:

   \[
   \binom{12}{4, 4, 4} = \frac{12!}{4! \times 4! \times 4!}
   \]

2. **Account for indistinguishability**:
   Since the groups are indistinguishable, we must divide by the number of ways to permute the three groups, which is \(3! = 6\). So, the number of ways to divide the children into three indistinguishable groups is:

   \[
   \frac{1}{3!} \times \binom{12}{4, 4, 4} = \frac{12!}{4! \times 4! \times 4! \times 3!}
   \]

3. **Calculate the value**:
   Let's calculate this expression.

\[
12! = 479,001,600
\quad \text{and} \quad
4! = 24
\quad \text{and} \quad
3! = 6
\]

Now, plug in the values:

\[
\frac{479,001,600}{24 \times 24 \times 24 \times 6} = \frac{479,001,600}{20,736}
\]

\[
\frac{479,001,600}{20,736} = 23,040
\]

Thus, the number of ways to divide the 12 children into three equal-sized, indistinguishable groups is **23,040**.

### Final Answer:
There are **23,040** ways to divide the 12 children into three equal-sized identical groups.

Would you like a further breakdown or additional details?

### Follow-up Questions:
1. How would the calculation change if the groups were distinguishable?
2. What would happen if the number of children was not divisible by 3?
3. How does the formula change for four groups instead of three?
4. Can this approach be generalized to unequal group sizes?
5. How would the result change if there were constraints on group membership?

### Tip:
For problems involving division into indistinguishable groups, always remember to account for group symmetry by dividing by the factorial of the number of groups!

In how many ways can 12 children be divided into three equal-sized identical (indistinguishable) groups?

Learn how to divide 12 children into three equal-sized, indistinguishable groups using combinatorial methods. This problem explores partitioning with the multinomial coefficient and symmetry adjustments. Ideal for Grades 10-12 students studying advanced combinatorics.

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