In a party every person shakes hands with every other person. If there are 105 hands shakes, find the number of person in the party.

Answer: Option A

Let n be the number of persons in the party
Number of hands shake = 105
Total number of hands shake is given by ⁿC₂
Now,
According to the question,
$$\begin{array}{rl}& {}^n{\text{C}}_2=105\\ & \text{or, }\frac{n!}{2!\times \left(n-2\right)!}=105\\ & \text{or, }\frac{n\times \left(n-1\right)}{2}=105\\ & \text{or, }{n}^2-n=210\\ & \text{or, }{n}^2-n-210=0\\ & \text{or, }n=15,-14\end{array}$$
But, we cannot take negative value of n
So, n = 15
i.e. number of persons in the party = 15