UK
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^@
- 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^@ - 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^@ - Hence, the number of persons at the party is ^@8^@.
