2022 AIME I Problems/Problem 9
Contents
Problem
Ellina has twelve blocks, two each of red (
), blue (
), yellow (
), green (
), orange (
), and purple (
). Call an arrangement of blocks 
if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement
![\[\textbf{R B B Y G G Y R O P P O}\]](http://latex.artofproblemsolving.com/2/2/4/2246d3f2384b544ed1634ec69c8d0c4be9f303e6.png)
is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is 
where 
and 
are relatively prime positive integers. Find 
Solution 1
Consider this position chart: ![\[\textbf{1 2 3 4 5 6 7 8 9 10 11 12}\]](http://latex.artofproblemsolving.com/d/2/6/d268e4b5ac272cbb26d410153e143b268e09cfa4.png)
Since there has to be an even number of spaces between each pair of the same color, spots 
, 
, 
, 
, 
, and 
contain some permutation of all 
colored balls. Likewise, so do the even spots, so the number of even configurations is 
(after putting every pair of colored balls in opposite parity positions, the configuration can be shown to be even). This is out of 
possible arrangements, so the probability is: ![\[\frac{6!\cdot6!}{\frac{12!}{(2!)^6}} = \frac{6!\cdot2^6}{7\cdot8\cdot9\cdot10\cdot11\cdot12} = \frac{2^4}{7\cdot11\cdot3} = \frac{16}{231},\]](http://latex.artofproblemsolving.com/a/9/9/a995946c0ecb2b47d5a183478a43f48d08e23fe6.png)
which is in simplest form. So, 
.
~Oxymoronic15
Solution 2
We can simply use constructive counting. First, let us place the red blocks; choose the first slot in 
ways, and the second in ways, because the number is cut in half due to the condition in the problem. This gives 
ways to place the red blocks. Similarly, there are 
ways to place the blue blocks, and so on, until there are 
ways to place the purple blocks. Thus, the probability is ![\[\frac{12 \cdot 6 \cdot 10 \cdot 5 \cdot 8 \cdot 4 \cdot 6 \cdot 3 \cdot 4 \cdot 2 \cdot 2 \cdot 1}{12!}=\frac{16}{231},\]](http://latex.artofproblemsolving.com/b/8/6/b8612c1f874b70bedcda99bfd7b81767c662389a.png)
and the desired answer is 
.
~A1001
Solution 3
Use constructive counting, as per above. WLOG, place the red blocks first. There are 11 ways to place them with distance 0, 9 ways them to place with distance 2, so on, so the way to place red blocks is 
. Then place any other block similarly, with 
ways (basic counting). You get then 
ways to place the blocks evenly, and 
ways to place the blocks in any way, so you get 
by simplifying.
-drag00n
Solution 4
We can divide the positions into odd and even positions. Each color needs one block in an odd position and one block in an even position.
WLOG, we place the first block of the first pair into an odd position. This leaves even positions out of the remaining positions for the second block of the first pair. This results in a probability of 
for the second block to fall into an even position.
We can now place the first block of the second pair into another odd position, leaving even positions out of the remaining positions for the second block of the second pair.
Continuing this pattern for the other 
pairs results in the product 
. Thus, our answer is .
~Zhixing
Solution 5
There are a total of 
ways to arrange the blocks because there are 
total ways to arrange the blocks but we're overcounting 
ways for each of the blocks. Now, to better visualize how we can arrange these blocks, consider the first slot. There are colors to choose from. WLOG, say it's red. Now, we can easily see that the next and last red block can be put into a total of 6 slots to make sure that the number of slots between the first slot and the next slot is even.
To better visualize this, I have drawn a chart type figure:
RR_ _ _ _ _ _ _ _ _ _
R _ _ R _ _ _ _ _ _ _ _
.
.
.
From this, you can get the idea. Counting up all these cases, it is obvious that there are ways to put the next and last red block. We continue by placing the next block into the next open slot. There are ways to choose the next color and similarly, only ways to put the next and last colored block to maintain an even number of slots. We get the idea. It seems that there are 
ways to put the blocks. We compute 
. This yields an answer of 
.
~ilikemath247365
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=dkoF7StwtrM
Video Solution (Power of Logic)
See Also
| 2022 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
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These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
