14 kids in the class, had to shake hands w/ eachother exactly once, how many handshakes occured?

Answer 1

Each of the 14 students shook hands with the 13 others (excluding themselves). But each handshake involved 2 students -- so one single handshake between A and B is counted by both.

If you asked each student for his tally of handshakes, you'd get 14 answers of 13 each. But (per above), that double-counts each handshake.

Thus, the answer is 14 * 13 / 2, which is 91.

Answer 2

there are a few ways to solve this.
hard way label each kid as a, b, c, d, ......
you can work this out as a shakes hand with 13 people
now b has already shaken hads with 12 people
now c has shaken hands with a, and b, so he shakes hands with 11 people
so the answer is 13+12+11+10+9+8+7+6+5+4+3+2+1

now a second way is to use factorials and combinations in which case the answer is
14!/2!/(14-2)!=14!/2!/12!=14*13/2!=14*13/2=7*13=91

Answer 3

c(14, 2) = 91