Let's break down the problem step by step to find the total number of handshakes that took place at the party.

Step 1: Total number of people

• There are 5 couples, so the total number of people is $5 \times 2 = 10$.

Step 2: Restrictions on handshakes

• Each person shakes hands with everyone else except:
  1. Their own partner.
  2. Themselves.

This means each person can only shake hands with $10 - 2 = 8$ other people.

Step 3: Calculating the total number of possible handshakes

Since each handshake involves two people, we must avoid double-counting. The number of distinct handshakes is given by the formula:

$\text{Total handshakes} = \frac{\text{number of people} \times (\text{number of people} - 1)}{2}$

Substituting the total number of people:

$\text{Total handshakes} = \frac{10 \times 9}{2} = 45$

Step 4: Jack's observation of 9 different answers

The problem mentions that Jack received 9 different answers, which might be related to the number of handshakes each person participated in. Since every individual has a different number of handshakes, ranging from 0 (if a person shook no hands) to 8 (if they shook hands with everyone except their partner), this fits the scenario.

Conclusion:

The total number of handshakes that took place at the party is 45.

Would you like more details or have any questions?

Related questions:

1. What is the handshake formula for any number of people?
2. If there were 6 couples, how many handshakes would occur?
3. Can you generalize this problem for $n$ couples?
4. What would happen if every person also shook hands with their partner?
5. How many handshakes occur if Jack didn't shake hands with anyone?

Tip:

For handshake problems, the formula $\frac{n(n-1)}{2}$ is useful to calculate combinations of 2 people from $n$ without repetition.