In how many ways can a pack of 52 cards be divided in 4 sets, three of them having 17 cards each and fourth just one card?

Answer

To solve the problem of dividing a pack of 52 cards into 4 sets, where three sets have 17 cards each and the fourth set has just one card, we can follow these steps:

### Step-by-Step Solution:

1. **Choose the single card**:
We start by selecting 1 card from the 52 cards to form the fourth set. The number of ways to choose 1 card from 52 is given by:
\[ \binom{52}{1} = 52 \]

2. **Choose the first set of 17 cards**:
After selecting the single card, we have 51 cards left. Now, we need to choose 17 cards for the first set. The number of ways to choose 17 cards from 51 is:
\[ \binom{51}{17} \]

3. **Choose the second set of 17 cards**:
After selecting the first set of 17 cards, we have 34 cards remaining (51 - 17 = 34). Now, we need to choose another 17 cards for the second set. The number of ways to choose 17 cards from 34 is:
\[ \binom{34}{17} \]

4. **Choose the third set of 17 cards**:
After selecting the second set of 17 cards, we have 17 cards left (34 - 17 = 17). We need to choose the last set of 17 cards, which will be the remaining cards. The number of ways to choose 17 cards from 17 is:
\[ \binom{17}{17} = 1 \]

5. **Account for the indistinguishable sets**:
Since the three sets of 17 cards are indistinguishable, we need to divide by the number of ways to arrange these three sets, which is \(3!\):
\[ 3! = 6 \]

6. **Combine all the choices**:
Now we can combine all these choices to get the total number of ways to divide the cards:
\[ \text{Total ways} = \frac{\binom{52}{1} \times \binom{51}{17} \times \binom{34}{17} \times \binom{17}{17}}{3!} \]

Substituting the values:
\[ \text{Total ways} = \frac{52 \times \binom{51}{17} \times \binom{34}{17} \times 1}{6} \]

### Final Calculation:
To find the total number of ways, we need to calculate the binomial coefficients:
- \(\binom{51}{17}\)
- \(\binom{34}{17}\)

After calculating these values and substituting them back into the equation, we can find the final answer.