arrangements of a platoon 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 ?

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.