If one person handshakes with the other only once and the number of handshakes is 66, then the number of persons will be

Question

Smart Achievers · Accepted Answer

(4)

Let the number of persons be $n$

One person out of $n$ persons will handshake with $(n - 1)$ persons.

Second person out of $(n - 1)$ persons will handshake with $(n - 2)$ persons.

Third person out of $(n - 2)$ persons will handshake with $(n - 3)$ persons.

$\begin{matrix} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \end{matrix}$

${(n - 1)}^{th}$ person out of 2 persons will handshake with 1 person.

So total number of handshakes

$= (n - 1) + (n - 2) + \dots + 2 + 1 = \frac{(n - 1) n}{2}$

$∴$ $\frac{n (n - 1)}{2} = 66$ $\Rightarrow n^{2} - n - 132 = 0$

$\Rightarrow (n - 12) (n + 11) = 0$ $\Rightarrow n = 12$ $(∵ n \neq - 11)$

So, Total number of persons = 12

Solution

Q.
If one person handshakes with the other only once and the number of handshakes is 66, then the number of persons will be

1 10

2 33

3 24

4 12

Want Us to Email you About Special Offers & Updates?