Question
Eight people are at a party. Everyone shakes hands once with everyone else. How many handshakes are there altogether?
Answers
GPT 3.5
To calculate the number of handshakes, we can use the combination formula.
Since each person shakes hands with every other person, each person will shake hands with 7 other people (since they don't shake hands with themselves).
So, for each person, there are 7 potential handshakes. However, this counts each handshake twice (once for each person involved in the handshake).
Therefore, we need to divide the total number by 2 to avoid counting each handshake twice.
Using the combination formula:
C(n, k) = n! / (k! * (n - k)!)
where n = 8 (the number of people) and k = 2 (the number of people involved in each handshake)
We can calculate the number of handshakes:
C(8, 2) = 8! / (2! * (8 - 2)!)
= 8! / (2! * 6!)
= (8 * 7 * 6!)/(2! * 6!)
= (8 * 7) / (2 * 1)
= 28
So, there are 28 handshakes altogether at the party.
Since each person shakes hands with every other person, each person will shake hands with 7 other people (since they don't shake hands with themselves).
So, for each person, there are 7 potential handshakes. However, this counts each handshake twice (once for each person involved in the handshake).
Therefore, we need to divide the total number by 2 to avoid counting each handshake twice.
Using the combination formula:
C(n, k) = n! / (k! * (n - k)!)
where n = 8 (the number of people) and k = 2 (the number of people involved in each handshake)
We can calculate the number of handshakes:
C(8, 2) = 8! / (2! * (8 - 2)!)
= 8! / (2! * 6!)
= (8 * 7 * 6!)/(2! * 6!)
= (8 * 7) / (2 * 1)
= 28
So, there are 28 handshakes altogether at the party.