In how many ways we can select four cards of different suits and different values from a deck of 52 cards?

Options

A. 10!
B. `(13!)/(10!)`
C. `(13!)/(9! )`
D. none of these

Answer

To solve the problem of selecting four cards of different suits and different values from a deck of 52 cards, we can follow these steps:

### Step 1: Understand the Structure of the Deck
A standard deck of 52 cards consists of 4 suits (Hearts, Diamonds, Clubs, Spades), and each suit has 13 cards of different values (Ace through King).

### Step 2: Selecting Cards from Different Suits
Since we need to select 4 cards, each from a different suit, we will select one card from each of the 4 suits.

### Step 3: Calculate the Number of Ways to Choose Cards
1. **Choose a card from the first suit**: There are 13 options (13C1 = 13).
2. **Choose a card from the second suit**: Again, there are 13 options (13C1 = 13).
3. **Choose a card from the third suit**: There are still 13 options (13C1 = 13).
4. **Choose a card from the fourth suit**: There are 13 options (13C1 = 13).

### Step 4: Total Combinations
Since we are selecting one card from each suit, we multiply the number of choices:
\[ \text{Total Ways} = 13 \times 13 \times 13 \times 13 = 13^4 \]

### Step 5: Calculate the Final Value
Calculating \( 13^4 \):
\[ 13^4 = 28561 \]

### Conclusion
Thus, the total number of ways to select four cards of different suits and different values from a deck of 52 cards is **28561**.

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