Question

A certain number of people attended a business party and each of them shakes hands with each other exactly once. if 190 handshakes happened in the party, how many attended the party?

Answer 1

190 handshakes,it means 1 handshake=2 people . therefore,190 handshakes=190×2 =380peoples

Answer 2

Answer:

20

Step-by-step explanation:

Let n be the no. of people in the party .

No. of people involved in 1 handshake = 2

So, we will use combination to find possible number of handshakes.

No. of handshakes = $\frac{n!}{r!(n-r)!}$

r = 2 and we are given that 190 handshakes happened in the party,

So,  $190=\frac{n!}{2!(n-2)!}$

$190=\frac{n(n-1)(n-2)!}{2!(n-2)!}$

$190=\frac{n(n-1)}{2 \times 1}$

$190 \times 2= n(n-1)$

$380=n^2-n$

$n^2-n-380=0$

$(n-20)(n+19)=0$

$n=20,-19$

Since no. of people cannot be negative

So, the no. of people in the party is 20