At a party, everyone must shake the hand of the other. If the total number of handshakes is 66, there are ----people at the party.
To find the number of people at the party, we can use the following formula for calculating the total number of handshakes in a group:
Total handshakes = (n * (n - 1)) / 2
Where "n" is the number of people at the party.
Given that there were 66 handshakes, we can set up the equation:
66 = (n * (n - 1)) / 2
Now, let's solve for "n." First, multiply both sides by 2 to remove the fraction:
2 * 66 = n * (n - 1)
132 = n^2 - n
Now, rearrange the equation:
n^2 - n - 132 = 0
We have a quadratic equation, and we can factor it:
(n - 12)(n + 11) = 0
Setting each factor equal to zero:
n - 12 = 0 n = 12
n + 11 = 0 n = -11
Since the number of people cannot be negative, we discard the negative solution.
Therefore, there were 12 people at the party.
The correct answer is 12.
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