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Ten men from a platoon are arranged in two rows. Each row has the men arranged by increasing height from left to right, and every man in the back row must be taller than the man in front of him.

In how many ways can the men be arranged in accordance with the conditions ?

It may not be difficult to get a solution by enumeration on this scale, but is there a neat way to solve the problem, and is it possible to generalise it for N men ?

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    I think you want to assume that no two men have the same height. Also, for generalizing to $N$, I can do the case where $N$ is prime.... Perhaps you mean for $ab$ men, in $a$ rows of $b$ men each.2012-01-16
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    @Gerry +1:-) I'm sure that the platoon commander would love to have a prime number of men in their prime. FYI, what usually happens is that the commander yells the number of rows, and the men need to sort it out. The last (and only the last) column may not be full of men. At least that's the way it worked, when I was doing my conscript duty. Mind you, at least in these parts the tallest dude would be on the right (when facing the commander) and on the left only from the commanders point of view. Either the latter is meant, or this is done differently wherever the OP is from.2012-01-16
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    @Jyrki, we seem to have come to the intersection of combinatorics and sociology, or perhaps combinatorics and military science. As a non-combatant, I yield to those with practical knowledge.2012-01-16

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The kind of arrangement you are asking about is called a "standard Young tableau," and you might find some answers by searching for that term.

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    Thanks a lot. I have found what i wanted in the "hook length" formula, http://mathworld.wolfram.com/HookLengthFormula.html One question, though. Why do you say you can generalise only for a prime N ?2012-01-16
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    What I said was that I could do the case where $N$ is prime, the idea being that the only rectangular array is $1\times N$, which is easy to work out. I made no assertion (in my comment) about other cases.2012-01-16