At a party, each person shakes hand with every other person. If | Math Question and Answer

Answer:

$8$

Step by Step Explanation:

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 $8$ .

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