In how many ways can 12 things be equally divided :
(i) between 2 persons (ii) into 2 heaps ?

Answer

To solve the problem of dividing 12 things equally, we will tackle both parts of the question step by step.

### Part (i): Dividing 12 things between 2 persons

1. **Understanding the Problem**: We need to divide 12 distinct items between 2 persons. Each person will receive 6 items.

2. **Choosing Items for One Person**: We can choose 6 items out of 12 for the first person. The number of ways to choose 6 items from 12 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 12 \) and \( r = 6 \):
\[ \binom{12}{6} = \frac{12!}{6!6!} \]

3. **Calculating the Value**:
\[ \binom{12}{6} = \frac{12!}{6!6!} \]
This can be simplified as:
\[ \binom{12}{6} = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 924 \]

4. **Considering the Two Persons**: Since the two persons are different, we do not need to divide by 2. Thus, the total number of ways to divide the items is:
\[ \text{Total ways} = 924 \]

### Part (ii): Dividing 12 things into 2 heaps

1. **Understanding the Problem**: Here, we need to divide 12 distinct items into 2 heaps. Each heap can have any number of items, including zero.

2. **Choosing Items for One Heap**: We can choose any number of items for one heap. If we choose \( k \) items for the first heap, the remaining \( 12 - k \) items will automatically go to the second heap.

3. **Calculating the Total Combinations**: The number of ways to choose \( k \) items from 12 is given by:
\[ \sum_{k=0}^{12} \binom{12}{k} \]
However, since the heaps are indistinguishable (i.e., Heap A and Heap B are the same as Heap B and Heap A), we need to divide by 2 to avoid double counting:
\[ \text{Total ways} = \frac{1}{2} \left( \sum_{k=0}^{12} \binom{12}{k} \right) \]

4. **Using the Binomial Theorem**: The sum of the binomial coefficients is given by:
\[ \sum_{k=0}^{n} \binom{n}{k} = 2^n \]
Therefore, for \( n = 12 \):
\[ \sum_{k=0}^{12} \binom{12}{k} = 2^{12} = 4096 \]

5. **Final Calculation**: Now we divide by 2:
\[ \text{Total ways} = \frac{4096}{2} = 2048 \]

### Final Answers:
(i) The number of ways to divide 12 things between 2 persons is **924**.
(ii) The number of ways to divide 12 things into 2 heaps is **2048**.