At a party, each person shakes hand with every other person. If there was a total of ^@28^@ handshakes, then how many persons were present there?


Answer:

^@8^@

Step by Step Explanation:
  1. Let the number of persons at the party be ^@n^@.
    Now, it is given that each person shakes hand with every other person in the party, therefore, the total number of handshakes ^@ = ^@ ^@^n C _2^@
  2. Also, we are given that the number of handshakes is ^@28^@, therefore, ^@^n C _2 = 28^@
    ^@\begin{align} & \implies \dfrac{ n! }{ 2! \times (n - 2)! } = 28 \\ & \implies \dfrac{ n(n - 1) }{2} = 28 \\ & \implies n^2 - n = 56 \\ & \implies n^2 - n - 56 = 0 \\ & \implies n^2 - 8 n + 7 n - 56 = 0 \\ & \implies n(n - 8) + 7(n - 8) = 0 \\ & \implies (n - 8)(n + 7) = 0 \\ & \implies n = 8 \space or \space n = -7 \end{align}^@
    Since, ^@n^@ cannot be negative, therefore ^@n = 8^@
  3. Hence, the number of persons at the party is ^@8^@.

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