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

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

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.