Question

If every person in a room shakes the right hand of every other person in the room, there are 36 possible handshakes. How many people are in the room?

Answer 1

There are `9` people in the room.

Explanation:

Suppose there are `n` people in the room.

If everyone shakes everyone else's hand there will be
`color(white)("XXX")nC_2 = (n!)/((n-2)!(2!)` handshakes.

We are told this number `=36`

`color(white)("XXX")(n!)/((n-2)!2!) = 36`

`rArrcolor(white)("XXX")nxx(n-1) = 36xx2!`

`rArrcolor(white)("XXX")n^2-n = 72`

`rArrcolor(white)("XXX")n^2-n-72=0`

`rArrcolor(white)("XXX")(n-9)(n+8)=0`

and since `n=-8` is a logical impossibility:

`color(white)("XXX")n=9`