Jack and his partner went to a party where four other couples were present. Every person shook hands with everyone else (other than their own partner). When the handshaking was over, Jack wanted to know how many handshakes took place. When he asked around, Jack got nine different answers.
How many handshakes took place?


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.

Jack and his partner went to a party where four other couples were present. Every person shook hands with everyone else (other than their own partner). When the handshaking was over, Jack wanted to know how many handshakes took place. When he asked around, Jack got nine different answers. How many handshakes took place?

Discover how to calculate the total number of handshakes at a party attended by 5 couples. Using combinatorics and the handshake formula n(n-1)/2, we solve the problem step by step, considering restrictions like not shaking hands with your partner. This problem is suitable for students in Grades 8-10.

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