You walk into a party without knowing anyone there. There are six women and four men and you know there are four married couples. In how many ways can you guess who the couples are? What if you know there are exactly three couples?

Let's break down both parts of the problem step by step:

### 1. **When there are exactly 4 married couples**:

There are 6 women and 4 men, and we know there are exactly 4 married couples. The task is to find how many ways you can guess which of the men are married to which women.

Here’s the reasoning:
- Out of the 6 women, we need to select 4 to be paired with the 4 men. The number of ways to choose 4 women from 6 is given by the combination formula \( \binom{6}{4} \). This represents selecting 4 women from the 6.
  
  \[
  \binom{6}{4} = \frac{6!}{4!(6-4)!} = 15
  \]

- Now, for each group of 4 women, there are \( 4! \) (4 factorial) ways to assign each of the 4 men to one of the 4 women. \( 4! = 24 \).

Thus, the total number of ways to guess the couples is:
\[
\text{Total ways} = \binom{6}{4} \times 4! = 15 \times 24 = 360
\]

### 2. **When there are exactly 3 married couples**:

Now, you know that there are exactly 3 married couples. The number of ways to guess the couples can be calculated similarly, but this time for only 3 pairs.

- Out of the 6 women, we need to select 3 to be paired with the 3 men. The number of ways to choose 3 women from 6 is given by the combination formula \( \binom{6}{3} \).

  \[
  \binom{6}{3} = \frac{6!}{3!(6-3)!} = 20
  \]

- For each group of 3 women, there are \( 3! \) (3 factorial) ways to assign each of the 3 men to one of the 3 women. \( 3! = 6 \).

Thus, the total number of ways to guess the couples is:
\[
\text{Total ways} = \binom{6}{3} \times 3! = 20 \times 6 = 120
\]

### Summary of results:

- **For exactly 4 married couples**: The number of ways to guess the couples is **360**.
- **For exactly 3 married couples**: The number of ways to guess the couples is **120**.

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Would you like more details on any part of the solution or have any questions?

Here are 5 questions to consider:
1. How does the combination formula work when selecting groups of women?
2. Why do we multiply by the factorial of men when assigning couples?
3. How would the problem change if there were 7 women and 5 men?
4. Can this method be generalized to other numbers of couples?
5. What would the answer be if no men or women were excluded from pairing?

**Tip**: Factorials grow very quickly, so pay attention to them in counting and probability problems!

Solve a combinatorics problem where you're asked to guess married couples at a party with 6 women and 4 men. Learn how to use combinations and permutations to calculate the number of ways to pair up 4 or 3 couples. Suitable for Grades 10-12.

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