Question

Everybody in a function shakes hands with everybody else. The total number of handshakes is 45. Find the number of people in the function.

Answer 1

We will assume the number of people in the function to be $x$. We will count the number of ways in which the handshakes are done so that everybody shakes hand with everybody else. To count this, we will use combinations. Each handshake involves 2 people, so we will choose 2 people out of the total number of people in the room and find the number of ways in which this can be done.

Complete step-by-step solution
Let the number of people in the function be $x$. We know that everybody in the function shakes hands with everybody else. For one handshake, there are two people involved. Now, we can choose the two people out of $x$ using a combination. The number of ways we can choose $r$ objects out of $n$ is given by
${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$
We can calculate the number of ways of choosing two people out of $x$ using the above formula. We already know that the total number of handshakes are 45. Therefore, we have ${}^{x}{{C}_{2}}=45$. We will substitute his value and $n=x$, $r=2$ in the above formula. We get the following equation,
$\begin{aligned} & {}^{x}{{C}_{2}}=\dfrac{x!}{2!\left( x-2 \right)!} \\\ & \Rightarrow 45=\dfrac{x\times \left( x-1 \right)\times \left( x-2 \right)!}{2!\left( x-2 \right)!} \\\ & \Rightarrow 45=\dfrac{x\times \left( x-1 \right)}{2} \\\ & \Rightarrow 90={{x}^{2}}-x \\\ & \therefore {{x}^{2}}-x-90=0 \\\ \end{aligned}$
We have obtained a quadratic equation in the variable $x$. We will solve the above quadratic equation using the method of factorization. We can write the middle term in the following manner,
${{x}^{2}}-10x+9x-90=0$
So, we can factorize the above equation as follows,
$\begin{aligned} & x\left( x-10 \right)+9\left( x-10 \right)=0 \\\ & \Rightarrow \left( x-10 \right)\left( x+9 \right)=0 \\\ & \therefore x=10\text{ or }x=-9 \\\ \end{aligned}$
As the number of people cannot be negative, we will discard the value of $x=-9$. Hence, we have $x=10$. The number of people in the function is 10.

Note: It is important to notice that we have to choose two people out of the total number of people in the room. We should be familiar with the working of a combination. We can solve the quadratic equation by other methods like using the quadratic formula according to convenience. It is useful to do the calculations explicitly so that minor mistakes can be avoided and we can obtain the correct answer.