n how many ways can be select 5 cards from a card pack such that all 4 suits appear?

To solve the problem of selecting 5 cards from a deck of 52 cards such that all four suits appear, we can break it down into a few steps:

### Step 1: Understand the problem
- There are 52 cards in a deck, divided into 4 suits: Spades, Hearts, Diamonds, and Clubs.
- We need to choose 5 cards such that all 4 suits are represented.

### Step 2: Consider the distribution of the suits
Since all four suits must be represented, the fifth card must be from one of the suits that have already appeared. The possible distributions of the 5 cards among the suits can be:
1. Four suits appear once each, and one suit appears twice.
  
### Step 3: Calculate the number of ways for each distribution

#### Case: Four suits appear once, one suit appears twice.
1. **Select the suit that will appear twice:**  
   There are 4 suits, so there are 4 choices for which suit appears twice.

2. **Select 2 cards from the chosen suit:**  
   There are 13 cards in each suit, so we need to choose 2 cards from the chosen suit. The number of ways to do this is \( \binom{13}{2} \).

3. **Select 1 card from each of the other 3 suits:**  
   For each of the other three suits, we need to select 1 card. The number of ways to do this is \( \binom{13}{1} = 13 \) for each suit. Since there are 3 suits, the number of ways to do this is \( 13 \times 13 \times 13 = 13^3 \).

### Step 4: Combine the results
Multiply the number of ways to choose the suit that appears twice, the number of ways to select 2 cards from that suit, and the number of ways to select 1 card from each of the other 3 suits.

\[
\text{Total number of ways} = 4 \times \binom{13}{2} \times 13^3
\]

### Step 5: Calculate the final answer
Let's compute the numerical value:

\[
\binom{13}{2} = \frac{13 \times 12}{2} = 78
\]
\[
\text{Total number of ways} = 4 \times 78 \times 13^3
\]
\[
13^3 = 2197
\]
\[
\text{Total number of ways} = 4 \times 78 \times 2197 = 685,464
\]

So, the total number of ways to select 5 cards such that all 4 suits appear is **685,464**.

Would you like more details on any part of this process, or do you have any further questions?

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Here are five related questions for further exploration:
1. How many ways can you select 5 cards such that they all belong to the same suit?
2. How many ways can you select 5 cards from a deck without any restriction on suits?
3. What is the probability of getting a hand with all 4 suits represented in a 5-card draw?
4. How many ways can you select 5 cards such that exactly 3 suits are represented?
5. How many ways can you select 5 cards such that no suit is repeated?

**Tip:** Remember that when dealing with problems involving card suits and combinations, considering the distribution of suits first can greatly simplify the calculations.

In how many ways can we select 5 cards from a card pack such that all 4 suits appear?

Learn how to calculate the total number of ways to select 5 cards from a deck such that all 4 suits appear. This detailed solution involves combinatorial techniques and covers essential concepts in probability and card games. Suitable for high school students in Grades 10-12.

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