Question

Everybody in a room shakes hands with everybody else. The total number of handshakes is $28$. Then, the total number of people in the room is
(A) $6$
(B) $8$
(C) $11$
(D) $9$

Answer 1

In the given question, we have to find the number of people in the room given the total number of handshakes. We will first discuss the logic and the concept of permutations and combinations that we will be using to find the answer to the given question.

Formula used:
We must remember the mathematical formula of combination to solve the problem as: $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$.

Complete step by step solution:
We are given that everybody in the room shakes hands with everybody else.
In order to count the total number of handshakes, we need to select and count pairs of people in the room so that a handshake is executed.
Let us assume the number of people in the room is $n$.
Then, we can select a pair of people in $^n{C_2}$ ways.
So, we get, $^n{C_2} = 28$
Expanding the combination formula, we get,
$\Rightarrow \dfrac{{n!}}{{2! \times \left( {n - 2} \right)!}} = 28$
Expanding the factorial of n, we get,
$\Rightarrow \dfrac{{n \times \left( {n - 1} \right) \times \left( {n - 2} \right)!}}{{2! \times \left( {n - 2} \right)!}} = 28$
Cancelling the common factors, we get,
$\Rightarrow \dfrac{{n \times \left( {n - 1} \right)}}{{2 \times 1}} = 28$
$\Rightarrow n\left( {n - 1} \right) = 56$
Simplifying the quadratic equation, we get,
$\Rightarrow {n^2} - n - 56 = 0$
Now, we solve the quadratic equation using splitting the middle term method.
We split the middle term $- n$ into two terms $- 8n$ and $7n$ since the product of these terms, $- 56{n^2}$ is equal to the product of the constant term and coefficient of ${x^2}$ and sum of these terms gives us the original middle term, $- n$.
So, we get,
$\Rightarrow {n^2} - 8n + 7n - 56 = 0$
Taking the common terms outside the bracket, we get,
$\Rightarrow n\left( {n - 8} \right) + 7\left( {n - 8} \right) = 0$
$\Rightarrow \left( {n + 7} \right)\left( {n - 8} \right) = 0$
Since the product of two terms is equal to zero, either of the two terms must be zero.
So, we get,
Either $\left( {n + 7} \right) = 0$ or $\left( {n - 8} \right) = 0$
Either $n = - 7$ or $n = 8$
We know that the value of $n$ cannot be negative. Hence, $n = 8$.
Therefore, there are 8 people in the room.

Note:
There are some constraints in the form of $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$ such as the value of $r$ cannot be negative and should be strictly less than the value of $n$. Also, we need to remember the fact that the notion of choosing $r$ objects out of $n$ objects is exactly equal to the notion of choosing $(n-r)$ objects out of n objects. The mathematical expression is $^n{C_{\left( {n - r} \right)}} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}{ = ^n}{C_r}$.