Difference between revisions of "2005 AIME I Problems/Problem 5"
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== Problem == | == Problem == | ||
| − | Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible | + | Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible [[distinguishable]] arrangements of the 8 coins. |
== Solution == | == Solution == | ||
| − | + | There are two separate parts to this problem: one is the color (gold vs silver), and the other is the orientation. | |
| + | There are <math>{8\choose4} = 70</math> ways to position the gold coins in the stack of 8 coins, which determines the positions of the silver coins. | ||
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| + | Create a string of letters H and T to denote the orientation of the top of the coin. To avoid making two faces touch, we cannot have the arrangement HT. Thus, all possible configurations must be a string of tails followed by a string of heads, since after the first H no more tails can appear. The first H can occur in a maximum of eight times different positions, and then there is also the possibility that it doesn’t occur at all, for <math>9</math> total configurations. Thus, the answer is <math>70 \cdot 9 = 630</math>. | ||
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== See also == | == See also == | ||
| − | + | {{AIME box|year=2005|n=I|num-b=4|num-a=6}} | |
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[[Category:Intermediate Combinatorics Problems]] | [[Category:Intermediate Combinatorics Problems]] | ||
Revision as of 14:36, 4 March 2007
Problem
Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.
Solution
There are two separate parts to this problem: one is the color (gold vs silver), and the other is the orientation.
There are 
ways to position the gold coins in the stack of 8 coins, which determines the positions of the silver coins.
Create a string of letters H and T to denote the orientation of the top of the coin. To avoid making two faces touch, we cannot have the arrangement HT. Thus, all possible configurations must be a string of tails followed by a string of heads, since after the first H no more tails can appear. The first H can occur in a maximum of eight times different positions, and then there is also the possibility that it doesn’t occur at all, for 
total configurations. Thus, the answer is 
.
See also
| 2005 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
