Question

At a party every person shakes hand every other person. If there was a total of $105$ handshakes at the party. Find the number of persons present at the party.

Answer 1

Hint: Think about that the number of people shakes hand how many times with each other, here we will use the formula of combination as the number of handshakes is ${}^n{{\text{C}}_2}$ .

Complete step by step solution:
Given: There are $105$ handshakes in the party when each person shakes hands with every other person then we have to find the number of persons present in the party.
Now, Let the number of person present in the party be $n$, then number of required persons are 2 to shake hands at one time, so number of handshakes will be a formula ${}^n{{\text{C}}_2}$ , we will put the number of handshakes equal to this to find the $n$ .
${}^n{{\text{C}}_2} = 105$.
Now, we will open the formula of combination which is written above.
$\dfrac{{n!}}{{2! \times \left( {n - 2} \right)!}} = 105$
Now, solve this equation further by doing multiplication and division.
$
  \dfrac{{n \times \left( {n - 1} \right) \times \left( {n - 2} \right)!}}{{2 \times \left( {n - 2} \right)!}} = 105 \\
  {n^2} - n = 210 \\
  {n^2} - n - 210 = 0 \\
$
Now, we have a quadratic equation and we will this equation with the splitting middle term method.
$
  {n^2} - \left( {15 - 14} \right)n - 210 = 0 \\
  {n^2} - 15n + 14n - 210 = 0 \\
  n\left( {n - 15} \right) + 14\left( {n - 15} \right) = 0 \\
  \left( {n - 15} \right)\left( {n + 14} \right) = 0 \\
$
Apply zero product rule on the obtained factors and solve for \[n\].
Thus,
$
  n - 15 = 0 \\
   \Rightarrow n = 15 \\
$ or $
  n + 14 = 0 \\
   \Rightarrow n = - 14 \\
$
So, $n$ will be 15 because if we put $\left( {n + 14} \right) = 0$ then the $n$ will be $ - 14$ and number of persons can never be negative so that is why $n$ will be 15.

$\therefore$ The number of persons present at the party is 15.

Note:
In the quadratic equation, the variable will always come with two solutions, after that we have to choose the solution according to the question if the number of something is asked or any integer.