Problem:
A game uses a deck of n different cards, where n is an integer and n≥6. The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find n.
Solution:
The conditions of the problem imply that (6n​)=6(3n​), so 6!(n−6)!n!​=6. 3!(n−3)!n!​. Then (n−6)!(n−3)!​=6 !, so (n−3)(n−4)(n−5)=720=10⋅9⋅8. Thus n=13 is a solution, and because (n−3)(n−4)(n−5) is increasing for n≥5, conclude that 13​ is the only solution for n≥5.
The problems on this page are the property of the MAA's American Mathematics Competitions