2005 AIME I Problems/Problem 5
Contents
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 1
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
.
Solution 2
We can imagine the coins as a string of
and
. Because no
adjacent coins can have
faces touching, subsequent to changing from
to
, the numbers following
must be
; therefore, the number of possible permutations if all the coins are indistinguishable is
(there are
possible places to change from
to
and there is the possibility that there no change occurs). There are
possibilities of what coins are gold and what coins are silver, so the solution 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 |
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