At a party, 55 handshakes took place as everyone shook hands with everyone else in attendance. How many people were at the party?
For n people the number of handshakes would be
C(n,2)
so
n!/(2!(n-2)!) = 55 , where n is a positive integer.
n(n-1)/2 = 55
n(n-1) = 110
n^2 - n - 110 = 0
(n-11)(n+10) = 0
n = 11
To find the number of people at the party, we can use a bit of logic and basic math. Let's assume there were n people at the party.
To shake hands with every other person, the first person will need to shake hands with n-1 people (as they cannot shake hands with themselves).
For the second person, they will need to shake hands with n-1 people as well, but they have already shaken hands with the first person. So, they need to shake hands with n-1-1 = n-2 people.
Following this pattern, the third person needs to shake hands with n-3 people, the fourth person needs to shake hands with n-4 people, and so on, until the last person who needs to shake hands with 2 people.
To find the total number of handshakes, we add up the number of handshakes each person needs to make:
(n-1) + (n-2) + (n-3) + ... + 2.
We know that 55 handshakes took place, so we can set up the equation:
(n-1) + (n-2) + (n-3) + ... + 2 = 55.
Now, we can solve this equation to find the value of n.