2017 AIME I Problems/Problem 11

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 $32\cdot720\cdot9$ , which is $207360$ . 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 $m=4$

1. There is a $\frac{15}{28}$ chance that exactly one of 1, 2, 3 is in the same row with 4.

There are 3 ways to select which of the smaller numbers will get in the row, and then 5

ways to select the number larger than 4.

$\frac{\dbinom{3}{1}\cdot\dbinom{5}{1}}{\dbinom{8}{2}} = \frac{15}{28}$

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

There are 2 ways to select the row that the two smaller number are in, and then $\dbinom{3}{2}$ ways

to place the smaller numbers in the row.

$\frac{\dbinom{2}{1}\cdot\dbinom{3}{2}}{\dbinom{6}{2}} = \frac{2}{5}$

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

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 permute the numbers in the individual Rows.

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

Solution 4

We see that if one of the medians is 5, then there are two remaining numbers greater than 5 and two less than 5, so it follows that $m=5$ . There are 3 ways to choose which row to have 5 in, $4\cdot 4=16$ ways to choose the other two numbers in that row, $3!=6$ ways to arrange the numbers in that row, and $6!=720$ ways for the remaining numbers, for our answer is $3\cdot 16\cdot 6\cdot 720=207\boxed{360}$ . -Stormersyle

2017 AIME I (Problems • Answer Key • Resources)
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