How many ways can 2 men and 2 women be selected for a debate tournament if there are 11 male finalists and 14 female finalists?

To solve this problem, we need to determine how many ways we can select 2 men out of 11 male finalists and 2 women out of 14 female finalists. This involves calculating the number of combinations for each gender separately and then multiplying the results.

The number of ways to select 2 men from 11 can be calculated using the combination formula:

\[
C(n, r) = \frac{n!}{r!(n-r)!}
\]

For the men:
- \( n = 11 \) (number of male finalists)
- \( r = 2 \) (number of men to select)

\[
C(11, 2) = \frac{11!}{2!(11-2)!} = \frac{11 \times 10}{2 \times 1} = 55
\]

For the women:
- \( n = 14 \) (number of female finalists)
- \( r = 2 \) (number of women to select)

\[
C(14, 2) = \frac{14!}{2!(14-2)!} = \frac{14 \times 13}{2 \times 1} = 91
\]

Now, to find the total number of ways to select 2 men and 2 women, multiply the combinations:

\[
\text{Total combinations} = C(11, 2) \times C(14, 2) = 55 \times 91 = 5005
\]

Thus, there are **5005 different ways** to select 2 men and 2 women for the debate tournament.

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What if we had to select 3 men and 3 women instead?
2. How would the solution change if there were only 10 male finalists?
3. What if the debate team required 2 men, 2 women, and 1 additional member of either gender?
4. How many ways can the selection be made if one specific man must be included?
5. What if the team selection was based on ranking rather than a random selection?

**Tip:** When solving combination problems, breaking the problem into smaller, independent parts (like selecting men and women separately) can simplify the calculations.

Learn how to calculate the number of ways to select 2 men and 2 women from 11 male finalists and 14 female finalists for a debate tournament. This problem involves combinatorics and the combination formula. Suitable for high school students (Grades 10-12).

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