At a party, there were 105 handshakes. If each person shook hands exactly once with every other person, how many people were at the party?
for n people there will be C(n,2) handshakes
C(n,2) = 105
n!/(2!(n-2)!) = 105
n(n-1)/2 = 105
n^2 - n - 210 = 0
(n-15(n+14) = 0
n = 15 or n=-14, but n > 0
so there were 15 people
Good
To find the number of people at the party, we can use a formula based on the concept of handshakes.
Let's assume that there are n people at the party. Each person shakes hands with every other person once, excluding themselves. So, person 1 shakes hands with n-1 people, person 2 shakes hands with n-1 people (excluding person 1), and so on.
If we sum up the number of handshakes each person makes, we should get the total number of handshakes at the party. In this case, we know that the total number of handshakes is 105.
So the equation would be: (n-1) + (n-1) + (n-1) + ... = 105
To solve this equation, we need to find the value of n. Let's simplify the equation:
(n-1) + (n-1) + (n-1) + ... = 105
n-1 + n-1 + n-1 + ... = 105
(n-1) * (n-1) = 105
n^2 - 2n + 1 = 105
n^2 - 2n - 104 = 0
Now, we can solve this quadratic equation to find the value of n. Using the quadratic formula:
n = (-(-2) ± √((-2)^2 - 4 * 1 * -104)) / (2 * 1)
n = (2 ± √(4 + 416)) / 2
n = (2 ± √420) / 2
n = (2 ± 20.49) / 2
Simplifying further:
n = (2 + 20.49) / 2 = 22.49/2 = 11.245
n = (2 - 20.49) / 2 = -18.49/2 = -9.245
Since the number of people cannot be negative, n must be 11.245, which is not a whole number. However, in this case, we are dealing with a real-life scenario, so the number of people at the party must be a whole number. Therefore, we need to round it up or down to the nearest whole number.
Rounding down, we have n = 11.
Hence, there were 11 people at the party.