Combination problem GRE.

Question

On new years Eve, every member of a community exchanged cards with every other member. If a total of 420 different cards were exchanged, then how many different members were there in the community? My answer is wrong. I did it in this way: There are five members: A B C D F; A=4, B=3, C=2, D=1, total number of cards exchanged =10.

Answer 1

Your method is wrong because the question says exchanged. That means that for each member will send cards to all remaining members. If there are five members A = 4, B=4, C =4 and D =4.

If there are x members, each will send x-1 cards.

So, equation for the case where 420 different cards were exchanged is

x.(x-1) = 420

Solving we get x=-20 or +21.

This means there were 21 members. (# of members cannot be -ve)