2005 AIME II Problems/Problem 1
Contents
[hide]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
Video Solution
https://youtu.be/IRyWOZQMTV8?t=150
~ pi_is_3.14
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. Since 720 is close to , we try 8, 9, and 10, which works, so and .
See Also
2005 AIME II (Problems • Answer Key • Resources) | ||
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Followed by Problem 2 | |
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