Create your permutation, combination, and probability distribution questions. One for each topic; in total three questions. Please add a table with your own. 

Here are three questions covering permutations, combinations, and probability distribution, including a table summarizing them:

| **Topic**                  | **Question**                                                                                             |
|----------------------------|----------------------------------------------------------------------------------------------------------|
| Permutations               | How many distinct ways can 5 students be seated in 5 chairs in a row?                                     |
| Combinations               | From a deck of 52 cards, how many different ways can you choose 3 cards from the set of 13 spades?        |
| Probability Distribution    | A fair 6-sided die is rolled twice. What is the probability distribution for the sum of the two rolls?    |

### 1. **Permutation Question:**
**Question:** How many distinct ways can 5 students be seated in 5 chairs in a row?

**Solution:**  
This is a permutation problem because the order matters. The number of ways to arrange 5 students in 5 chairs is given by \( 5! \) (5 factorial):

\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

Thus, there are **120 distinct seating arrangements**.

---

### 2. **Combination Question:**
**Question:** From a deck of 52 cards, how many different ways can you choose 3 cards from the set of 13 spades?

**Solution:**  
This is a combination problem since the order of the cards doesn’t matter. The number of ways to choose 3 cards from the 13 spades is given by the combination formula \( \binom{n}{r} \), where \( n = 13 \) and \( r = 3 \):

\[
\binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286
\]

So, there are **286 ways** to choose 3 spades.

---

### 3. **Probability Distribution Question:**
**Question:** A fair 6-sided die is rolled twice. What is the probability distribution for the sum of the two rolls?

**Solution:**
We calculate the sum of the two rolls, which can range from 2 (1+1) to 12 (6+6). There are 36 possible outcomes (since \( 6 \times 6 = 36 \)). The probabilities for each sum can be calculated by counting how many ways each sum can occur.

| **Sum** | **Number of Ways** | **Probability** |
|---------|--------------------|-----------------|
| 2       | 1 (1+1)            | \( \frac{1}{36} \) |
| 3       | 2 (1+2, 2+1)       | \( \frac{2}{36} \) |
| 4       | 3 (1+3, 2+2, 3+1)  | \( \frac{3}{36} \) |
| 5       | 4 (1+4, 2+3, 3+2, 4+1) | \( \frac{4}{36} \) |
| 6       | 5 (1+5, 2+4, 3+3, 4+2, 5+1) | \( \frac{5}{36} \) |
| 7       | 6 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1) | \( \frac{6}{36} \) |
| 8       | 5 (2+6, 3+5, 4+4, 5+3, 6+2) | \( \frac{5}{36} \) |
| 9       | 4 (3+6, 4+5, 5+4, 6+3) | \( \frac{4}{36} \) |
| 10      | 3 (4+6, 5+5, 6+4) | \( \frac{3}{36} \) |
| 11      | 2 (5+6, 6+5) | \( \frac{2}{36} \) |
| 12      | 1 (6+6) | \( \frac{1}{36} \) |

So, the probability distribution for the sum of the two rolls is as shown above.

---

Would you like more details or explanations for any of these questions? Here are five related questions to consider:

1. How would the seating arrangement change if some students refuse to sit next to each other?
2. What if you choose 5 cards from the 13 spades? How many combinations are possible?
3. What is the expected value of the sum in the probability distribution for two dice?
4. How does the probability distribution change if one of the dice is biased?
5. How do you calculate permutations if some items are repeated?

**Tip:** Always distinguish between permutations (where order matters) and combinations (where order doesn't) when solving problems.

Create your permutation, combination, and probability distribution questions. One for each topic; in total three questions. Please add a table with your own.

Explore three questions covering permutations, combinations, and probability distribution, complete with solutions. Learn how to calculate permutations, combinations from a deck of cards, and the probability distribution for the sum of two dice rolls. Ideal for students in Grades 9-12.

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