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A certain platoon is made up of 3 squads, each of which has 4 soldiers. When the platoon lines up to enter the mess hall, the squads are allowed to be in any order but the soldiers must line up within their squads according to certain rules. The soldiers in the first squad can line up any way they want as long as they stay with their squad. The squad leader of the second squad insists that the soldiers in that squad be in one particular order. the third squad leader wants the soldiers in that squad to line up in order from either tallest to shortest or shortest to tallest. How many different ways can the platoon line up?
Answer is 288.
This question involves the "groups of groups" pattern. First consider how many ways the groups (squads) can be arranged. There are 3 distinct squads, so there are 3! = 3 x 2 x 1 = 6 different ways. For the squad that is permitted to choose any order they wish, there are 4! = 4 x 3 x 2 x 1 = 24 different ways they can line up. The squad that lines up by height can only have 2 variations and the remaining squad only has one way to line up within the squad. Therefore, the total number of ways that the platoon can line up is 6 x 24 x 2 x 1 = 288.
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