At a party everyone shook hands with everyone exactly once there were a total of 36 handshakes how many people were at t he party
combinations of n people r at a time
= n!/[ r! (n-r)! ]
n = ?
r = 2
36 = n!/[ 2 (n-2)! ]
72 = n! /(n-2)!
72 = n (n-1) the rest cancels
72 = n^2 -n
n^2 - n - 72 = 0
(n-9)(n+8) = 0
n = 9
combinations of n people r at a time
= n!/[ r! (n-r)! ]
n = ?
r = 2
36 = n!/[ 2 (n-2)! ]
72 = n! /(n-2)!
72 = n (n-1) the rest cancels
72 = n^2 -n
n^2 - n - 72 = 0
(n-9)(n+8) = 0
n = 9
To find out how many people were at the party, we can use a simple formula.
Let's assume the number of people at the party is 'n'.
In a group where everyone shakes hands with everyone exactly once, each person shakes hands with (n-1) other people (since they shake hands with everyone except themselves).
Now, if there are 'n' people at the party, the total number of handshakes can be calculated by summing up the number of handshakes each person makes.
Since each person shakes hands with (n-1) other people, the total number of handshakes is n * (n-1).
According to the problem, the total number of handshakes is 36. Therefore, we can set up the equation as follows:
n * (n-1) = 36
Now, we can solve this equation to find the value of 'n'.
Let's solve it step by step:
n^2 - n = 36 (expand the expression)
n^2 - n - 36 = 0 (rearrange the terms)
Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula.
By factoring, we can rewrite the equation as:
(n-6)(n+5) = 0
This means that either (n-6) = 0 or (n+5) = 0.
So, we have two possible solutions: n = 6 or n = -5.
Since the number of people cannot be negative, we disregard the n = -5 solution.
Therefore, the number of people at the party is n = 6.
Hence, there were 6 people at the party.