1. ## Handshakes

At a picnic there were c children, f adult females, and m adult males, where 2<c<f<m. Each person shook hands with every other person. The sum of the number of handshakes among children, the number of handshakes among adult females, and the number of handshakes among adult males is 57. How many handshakes were there altogether?

2. Hello, MathMage89!

An unusual problem . . . and I have an unusual solution.

At a picnic there were c children, f adult females, and m adult males,
. . where 2 < c < f < m.
Each person shook hands with every other person.
The sum of the number of handshakes among children,
. . the number of handshakes among adult females,
. . and the number of handshakes among adult males is 57.
How many handshakes were there altogether?

Consider the number of handshakes among n children.

2 children: .C(2,2) = 1
3 children: .C(3,2) = 3
4 children: .C(4,2) = 6
5 children: .C(5,2) = 10 . . . etc.

You may recognize these as "triangular" numbers.

The triangular numbers less than 57 are: .3, 6, 10, 15, 21, 28, 36, 45, 55.
. . (The "1" is discarded; there were at least 2 children.)

The only three with a sum of 57 is: .6 + 15 + 36

Hence, there were 3 children, 5 adult females, and 9 adult males.

With 17 people, there were C(17,2) = 136 handshakes.