How many handshakes in this room?

How did you make it 'select one or more' type?

Each of the $100$ people shakes hands with $99$ others. If we total these, we get $9900$ ; but we've counted every handshake twice! So the number of handshakes is $\boxed{4950}$ .

(Another way of looking at this is that there are $9900$ "person-handshakes", each of which involves two people)

• The first person will shake the hands of the other 99 people.
• Because they have shaken the hand of the first person, the next person will, therefore, shake the hands of 98 people.
• The third will shake 97 hands, and so on and so on.
• This would suggest that the total number of handshakes would equal; 99+98+97+...+1

To save time, this can be simplified to; $100*(\frac{99}{2}$ ) = 4950 handshakes.

$100\choose 2$ =4950

OR

$\frac{n(n-1)}{2}=\frac{100\times 99}{2}=4950$

It is a combination problem and you can use the formula nC2.

Each of the 100 members will shake the hand of 99 people (excluding himself). Every handshake is between two people and thus counted twice. So we get:

$\frac{100\cdot99}{2} = \boxed{4950}$