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Question
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.
18
9
12
15
none of these
ANSWER : 3
Descrption
<p style="margin-left:0px;">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:</p><p style="margin-left:0px;">Total handshakes = (n * (n - 1)) / 2</p><p style="margin-left:0px;">Where "n" is the number of people at the party.</p><p style="margin-left:0px;">Given that there were 66 handshakes, we can set up the equation:</p><p style="margin-left:0px;">66 = (n * (n - 1)) / 2</p><p style="margin-left:0px;">Now, let's solve for "n." First, multiply both sides by 2 to remove the fraction:</p><p style="margin-left:0px;">2 * 66 = n * (n - 1)</p><p style="margin-left:0px;">132 = n^2 - n</p><p style="margin-left:0px;">Now, rearrange the equation:</p><p style="margin-left:0px;">n^2 - n - 132 = 0</p><p style="margin-left:0px;">We have a quadratic equation, and we can factor it:</p><p style="margin-left:0px;">(n - 12)(n + 11) = 0</p><p style="margin-left:0px;">Setting each factor equal to zero:</p><p style="margin-left:0px;">n - 12 = 0 n = 12</p><p style="margin-left:0px;">n + 11 = 0 n = -11</p><p style="margin-left:0px;">Since the number of people cannot be negative, we discard the negative solution.</p><p style="margin-left:0px;">Therefore, there were 12 people at the party.</p><p style="margin-left:0px;">The correct answer is 12.</p>
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