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Revision as of 13:31, 14 October 2017

Problem 11

Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$.

Solution 1

We know that if $5$ is a median, then $5$ will be the median of the medians.

WLOG, assume $5$ is in the upper left corner. One of the two other values in the top row needs to be below $5$, and the other needs to be above $5$. This can be done in $4\cdot4\cdot2=32$ ways. The other $6$ can be arranged in $6!=720$ ways. Finally, accounting for when $5$ is in every other space, our answer is $32\cdot720\cdot9$. But we only need the last $3$ digits, so $\boxed{360}$ is our answer.

~Solution by SuperSaiyanOver9000, mathics42


Solution 2

(Complementary Counting with probability)

Notice that m can only equal 4, 5, or 6, and 4 and 6 are symmetric.

WLOG let $m=4$

There is a $\frac{15}{28}$ chance that exactly one of 1, 2, 3 is in the same row.

There is a $\frac{2}{5}$ chance that the other two smaller numbers end up in the same row.

$9!(1-2*\frac{15}{28}*\frac{2}{5})=362880*\frac{4}{7}=207\boxed{360}$.


Solution 3

We will make sure to multiply by $3!$ in the end to account for all the possible permutation of the rows.


WLOG, let $5$ be present in the Row #$1$.


Notice that $5$ MUST be placed with a number lower than it and a number higher than it.


This happens in $4\cdot4$ ways. You can permutate Row #$1$ in $3!$ ways.


Now, take a look at Row $2$ and Row $3$.


Because there are $6$ numbers to choose from now, you can assign #'s to Row's #2&3 in


$\frac{\binom{6}{3}\cdot\binom{3}{3}}{2}$ ways. There are $3!\cdot3!$ ways to permutate the numbers in the individual Rows.

Hence, our answer is $3!(4\cdot4\cdot3!\cdot{10}\cdot{3!}\cdot{3!}=3!(16\cdot60\cdot36)=3!(34560)\implies{207\boxed{360}}$

See Also

2017 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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