Question

At a party, everyone shook hands with everyone else. It was found that 66 handshakes were exchanged. The number of persons who attended the party was
A. 10
B. 12
C. 19
D. 20

Answer 1

Hint: At first we have to assume the total number of persons who attended the party was n. Then we have to clear the idea that if everyone shakes hands with everyone else, then the first person shakes hands (n – 1) times. By this concept we will solve the problem.

Complete step by step answer:
To clear out the idea let us assume there are three persons A, B, C who shook hands with each other. So, now A shook hands with B and C, B shook hands with A and C. Therefore, if there are three people then the first person shakes hand (3 – 1) times i.e. 2 times and the second person shakes hand (3 – 2) time i.e. 1 time. So, if we assume in this problem that there are no persons. Then, the first person shakes hand with everyone else (n – 1). Second person shakes hands with everyone else (n – 2). Third person shakes hands with everyone else (n – 3) and so on.
So, total handshakes will be = (n – 1) + (n – 2) + (n – 3) + … + 0
= \[\dfrac{{(n - 1)(n - 1 + 1)}}{2} = \dfrac{{n(n - 1)}}{2}\]
[Since, 1 + 2 + 3 + 4 +… + n = $\dfrac{{n(n + 1)}}{2}$ ]
And also given that total number of handshakes = 66.
According to the given question,
$ \dfrac{{n(n - 1)}}{2} = 66 \\$
$ \Rightarrow {n^2} - n = 132 \\$
On simplifying the above equation, we get
$ \Rightarrow {n^2} - n - 132 = 0 \\ $
Solve the quadratic equation, by middle term splitting method
$ \Rightarrow {n^2} - 12n + 11n - 132 = 0 \\$
On simplifying the above equation, we get
$\Rightarrow n(n - 12) + 11(n - 12) = 0 \\$
On further simplification,
$\Rightarrow (n - 12)(n + 11) = 0 \\$
$\Rightarrow n = 12 or n=12 \\ $
We do not consider $n = -11$ as $n$ cannot be negative.

$\therefore$The number of persons who attended the party is 12. Hence, option (B) is correct.

Note:
To solve these types of questions, we should have a clear image of the problem by assuming the problem in small statements. While doing the calculations we need to take proper care.