Difference between revisions of "2005 AIME II Problems/Problem 1"
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[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] | ||
Revision as of 20:53, 7 September 2006
Problem
A game uses a deck of 
different cards, where is an integer and 
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 
Solution
The number of ways to draw six cards from is given by the binomial coefficient 
. The number of ways to choose three cards from is 
. We are given that 
, so 
. Cancelling like terms, we get 
. We must find a factorization of the left-hand side of this equation into three consecutive integers. With a little work we realize the factorization 
, so 
and 
