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Let's go step by step and solve both questions based on the given information.

**1. If the cast consists of 2 women and 2 men, how many different casts can be selected?**

To solve this, we need to select 2 women from a group of 9 women and 2 men from a group of 5 men. The number of ways to select 2 people from a group is given by combinations, denoted as \( \binom{n}{r} \), where \( n \) is the total number of people and \( r \) is the number of people to select.

- For women: \( \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \)
- For men: \( \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \)

Now, multiply the two results to find the total number of different casts:

\[
\text{Total casts} = \binom{9}{2} \times \binom{5}{2} = 36 \times 10 = 360
\]

**Answer for Question 1: 360 casts**

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**2. If the cast consists of 2 women or 2 men, how many different casts can be selected?**

In this case, we have two separate cases: selecting only 2 women or selecting only 2 men. We need to calculate both cases and then add them, since "or" in probability means addition.

- For 2 women: \( \binom{9}{2} = 36 \)
- For 2 men: \( \binom{5}{2} = 10 \)

Adding the two results:

\[
\text{Total casts} = \binom{9}{2} + \binom{5}{2} = 36 + 10 = 46
\]

**Answer for Question 2: 46 casts**

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Would you like further clarification on any step? Here are 5 related questions you might be interested in:

1. How are combinations different from permutations?
2. How do you calculate combinations for larger groups?
3. What if the cast required 3 women and 2 men? How would you approach that?
4. How do combinations change when there's no gender restriction?
5. What if the requirement was 2 women *and/or* 2 men, including mixed genders?

**Tip:** When using combinations, remember that order does not matter in the selection.

There are 5 men and 9 women auditioning to be in the school play. If the cast consists of 2 women and 2 men, how many different casts can be selected? If the cast consists of 2 women or 2 men, how many different casts can be selected?

Learn how to calculate different ways to select 2 women and 2 men from a group of 5 men and 9 women using combinations. This problem demonstrates key concepts in combinatorics, perfect for students in Grades 9-11.

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