There are `n` married couples at a party. Each person shakes hand with every person other than her or his spouse. Find the total m of hand shakes.

Options

A. `""^(2n)C_(2)-n`
B. `""^(2n)C_(2)-(n-1)`
C. `2n(n-1)`
D. `""^(2n)C_(2)`

Answer

To solve the problem of finding the total number of handshakes at a party with `n` married couples, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Total Number of People**:
Since there are `n` married couples, the total number of people at the party is:
\[ \text{Total people} = 2n \]

2. **Calculate Total Handshakes Without Restrictions**:
If there were no restrictions (i.e., if everyone could shake hands with everyone else), the total number of ways to choose 2 people from `2n` people is given by the combination formula:
\[ \text{Total handshakes} = \binom{2n}{2} = \frac{2n(2n-1)}{2} = n(2n-1) \]

3. **Account for the Restriction (No Handshakes with Spouse)**:
Each person does not shake hands with their spouse. Since there are `n` couples, there are `n` handshakes that do not occur (one for each couple). Therefore, we need to subtract these `n` handshakes from the total calculated in the previous step:
\[ \text{Valid handshakes} = n(2n-1) - n \]

4. **Simplify the Expression**:
Now, simplify the expression:
\[ \text{Valid handshakes} = n(2n-1) - n = n(2n-1 - 1) = n(2n - 2) = 2n(n - 1) \]

5. **Final Result**:
Thus, the total number of valid handshakes at the party is:
\[ m = 2n(n - 1) \]

### Summary
The total number of handshakes at the party, where no one shakes hands with their spouse, is given by the formula:
\[ m = 2n(n - 1) \]