In how many ways can a pack of 52 cards be formed into 4 groups of 13 cards each

Answer

To find the number of ways to form a pack of 52 cards into 4 groups of 13 cards each, we can follow these steps:

### Step 1: Understand the Problem
We need to divide 52 cards into 4 groups, where each group contains exactly 13 cards. The order of the groups does not matter, which means that the groups are indistinguishable.

### Step 2: Calculate the Combinations
To solve this, we can use the concept of combinations. The number of ways to choose 13 cards from 52 is given by the combination formula \( \binom{n}{r} \), which is defined as:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

### Step 3: Apply the Combination Formula
1. **Choose the first group (Group A)**: The number of ways to choose 13 cards from 52 is:
\[ \binom{52}{13} \]

2. **Choose the second group (Group B)**: After choosing Group A, there are 39 cards left. The number of ways to choose 13 cards from these 39 is:
\[ \binom{39}{13} \]

3. **Choose the third group (Group C)**: After choosing Groups A and B, there are 26 cards left. The number of ways to choose 13 cards from these 26 is:
\[ \binom{26}{13} \]

4. **Choose the fourth group (Group D)**: Finally, the last group will automatically consist of the remaining 13 cards. The number of ways to choose 13 cards from 13 is:
\[ \binom{13}{13} = 1 \]

### Step 4: Combine the Results
The total number of ways to form the groups 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 5: Adjust for Indistinguishable Groups
Since the groups are indistinguishable, we must divide by the number of ways to arrange the 4 groups, which is \(4!\):
\[ \text{Final Result} = \frac{\binom{52}{13} \times \binom{39}{13} \times \binom{26}{13} \times \binom{13}{13}}{4!} \]

### Step 6: Substitute Values and Simplify
Now we can substitute the values into the formula and simplify:
\[ \text{Final Result} = \frac{\binom{52}{13} \times \binom{39}{13} \times \binom{26}{13} \times 1}{4!} \]

### Final Answer
Thus, the number of ways to form a pack of 52 cards into 4 groups of 13 cards each is:
\[ \frac{\binom{52}{13} \times \binom{39}{13} \times \binom{26}{13}}{4!} \]