jmc

Question

Twenty gremlins and fifteen imps are at the Annual Mischief Convention. The imps have had a lot of in-fighting lately and refuse to shake hands with each other, but they readily shake hands with all of the gremlins. Meanwhile, all the gremlins are quite friendly and shake hands with all of the other gremlins as well as imps. Each pair of creatures shakes hands at most once. How many handshakes were at the convention?

Accepted Answer

Let us first count the handshakes between two gremlins. There are

20

gremlins, so there must be

\frac{20 \cdot 19}{2} = 190

handshakes between two gremlins, making sure to divide by two to avoid over-counting.

Meanwhile, there are

15

imps that shake hands with each of the

20

gremlins, which makes

15 \cdot 20 = 300

handshakes between imps and gremlins.

Adding them up, we have

300 + 190 = 490

total handshakes.