In how many ways one can make four equal heaps using a pack of 52 playing cards?

Answer

To solve the problem of distributing 52 playing cards into four equal heaps, we need to follow these steps:

### Step 1: Understand the Distribution
We have 52 cards and we want to divide them into four equal heaps. Each heap will contain \( \frac{52}{4} = 13 \) cards.

### Step 2: Choose Cards for Each Heap
1. **Choose cards for the first heap:** We can select 13 cards from the 52 cards. The number of ways to do this is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. Thus, the number of ways to choose the first heap is:
\[ \binom{52}{13} \]

2. **Choose cards for the second heap:** After selecting the first heap, we have 39 cards left. We now choose 13 cards from these 39:
\[ \binom{39}{13} \]

3. **Choose cards for the third heap:** Now, we have 26 cards remaining. We choose 13 cards from these 26:
\[ \binom{26}{13} \]

4. **Choose cards for the fourth heap:** Finally, we take the remaining 13 cards for the fourth heap. There is only one way to do this:
\[ \binom{13}{13} = 1 \]

### Step 3: Calculate the Total Arrangements
The total number of ways to distribute the cards into the four heaps is the product of the combinations calculated above:
\[ \text{Total Ways} = \binom{52}{13} \times \binom{39}{13} \times \binom{26}{13} \times \binom{13}{13} \]

### Step 4: Adjust for Indistinguishable Heaps
Since the heaps are indistinguishable (the order of heaps does not matter), we need to divide by the number of ways to arrange the four heaps, which is \( 4! \):
\[ \text{Final Count} = \frac{\binom{52}{13} \times \binom{39}{13} \times \binom{26}{13} \times \binom{13}{13}}{4!} \]

### Step 5: Simplify the Expression
Using the factorial representation, we can express the combinations as:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Thus, we can write:
\[ \text{Final Count} = \frac{52!}{13! \cdot 39!} \times \frac{39!}{13! \cdot 26!} \times \frac{26!}{13! \cdot 13!} \times 1 \div 4! \]

This simplifies to:
\[ \text{Final Count} = \frac{52!}{(13!)^4 \cdot 4!} \]

### Conclusion
This expression gives us the total number of ways to create four equal heaps from a pack of 52 playing cards.

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