Hello, MathMage89!

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

At a picnic there werecchildren,fadult females, andmadult 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 amongnchildren.

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.