In how many ways can you select 3 different numbers from the set 1 2 3 4 5 6 7 8 so that the numbers could represent the measures of the sides of a triangle?

A triangle can be constructed provided the combined length of the shortest two sides is greater than the longest side. Clearly a side of length 8 cannot be one of the two smaller sides. Also, a side of length 1 cannot be used. (Here's why: suppose the shortest side is of length 1, and the next shortest of length N. In this case the longest side must be smaller than N+1 but - by definition - greater than N, which is impossible if we are to use only unique integer values.) Therefore the shorter two sides must take values from 2 to 7. The possible combinations of shorter sides are listed below, alongside feasible values for the third side: * [short side 1, short side 2 - possible longest sides] * 2,3 - 4 * 2,4 - 5 * 2,5 - 6 * 2,6 - 7 * 2,7 - 8 * 3,4 - 5,6 * 3,5 - 6,7 * 3,6 - 7,8 * 3,7 - 8 * 4,5 - 6,7,8 * 4,6 - 7,8 * 4,7 - 8 * 5,6 - 7,8 * 5,7 - 8 * 6,7- 8 Count all those possibilities up and you will find a total of 22 unique solutions.