Difference between revisions of "2005 AIME II Problems/Problem 1"
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A game uses a deck of <math> n </math> different cards, where <math> n </math> is an integer and <math> n \geq 6. </math> 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 <math> n. </math> | A game uses a deck of <math> n </math> different cards, where <math> n </math> is an integer and <math> n \geq 6. </math> 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 <math> n. </math> | ||
| + | |||
| + | == Video Solution == | ||
| + | https://youtu.be/IRyWOZQMTV8?t=150 | ||
| + | |||
| + | ~ pi_is_3.14 | ||
== Solution == | == Solution == | ||
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We are given that <math>{n\choose 6} = 6 {n \choose 3}</math>, so <math>\frac{n\cdot(n-1)\cdot(n-2)\cdot(n-3)\cdot(n-4)\cdot(n-5)}{6\cdot5\cdot4\cdot3\cdot2\cdot1} = 6 \frac{n\cdot(n-1)\cdot(n-2)}{3\cdot2\cdot1}</math>. | We are given that <math>{n\choose 6} = 6 {n \choose 3}</math>, so <math>\frac{n\cdot(n-1)\cdot(n-2)\cdot(n-3)\cdot(n-4)\cdot(n-5)}{6\cdot5\cdot4\cdot3\cdot2\cdot1} = 6 \frac{n\cdot(n-1)\cdot(n-2)}{3\cdot2\cdot1}</math>. | ||
| − | Cancelling like terms, we get <math>(n - 3)(n - 4)(n - 5) = | + | Cancelling like terms, we get <math>(n - 3)(n - 4)(n - 5) = 720</math>. |
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| − | + | We must find a [[factoring|factorization]] of the left-hand side of this equation into three consecutive [[integer]]s. Since 720 is close to <math>9^3=729</math>, we try 8, 9, and 10, which works, so <math>n - 3 = 10</math> and <math>n = \boxed{13}</math>. | |
== See Also == | == See Also == | ||
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[[Category:Introductory Combinatorics Problems]] | [[Category:Introductory Combinatorics Problems]] | ||
| + | [[Category:Introductory Algebra Problems]] | ||
| + | |||
| + | {{MAA Notice}} | ||
Latest revision as of 19:48, 31 May 2025
Contents
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) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
