Question 1092064: In how many ways can 3 americans,4 Frenchmen, 4 danes and 2 italians be seated in a row so that those of the same nationality sit together
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website! Since the people of the same nationality will be sitting together, this problem is identical to one where there is one person of each of 4 different nationalities and we need to seat them into 4 seats. Let's label the nationalities A,B,C,D
The seats can be drawn __1__|__2__|__3__|__4__ (if you must, you can think of each numbered seat as a GROUP of seats which will hold all the people of a given nationality).
There are 4 choices (A,B,C,D) to sit in seat 1
That leaves 3 choices for seat 2
2 choices for seat 3
1 choice for seat 4
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We must multiply the above choices, because, for example, after choosing who sits in seat 1, the other 3 seats are chosen independently (say B sits in seat 1, then A,C,and D can shuffle themselves in the remaining 3 seats any way they want). Mathematically, it is written 4! = 4*3*2*1 = 
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