Difference between revisions of "2005 AIME II Problems/Problem 1"
m |
|||
| Line 9: | Line 9: | ||
== See Also == | == See Also == | ||
| + | *[[2005 AIME II Problems/Problem 2| Next problem]] | ||
*[[2005 AIME II Problems]] | *[[2005 AIME II Problems]] | ||
[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] | ||
Revision as of 19:53, 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 
