In a party every person shakes hand with every other person. If there was a total 105 handshakes in the party. Find the number of people present who were present in the party?

Question

ganeshie8 · Answer

Say there are $n$ people in the room. 

The first person may shake hands with $n-1$ other people.

The next person may shake hand with $n-2$ other people, not counting the first person again.

... 

Adding them up gives
$$(n-1)+(n-2)+\cdots+3+2+1$$

which is same as the sum of first $n-1$ natural numbers.

ganeshie8 · Answer

That means we end up solving a quadratic equation
$$\dfrac{n(n-1)}{2} = 105$$

anonymous · Answer

hello

anonymous · Answer

i have also got another method,simply we will apply combination NC2
$$N!/2 !(N-2) !=105$$

anonymous · Answer

anyhow thanks for the help!

ganeshie8 · Answer

Yes I like your method