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# The 12 workers in a cafeteria crew rotate among three kinds of jobs. In how many ways can the crew?

## The 12 workers in a cafeteria crew rotate among three kinds of jobs. In how many ways can the crew be assigned the jobs of 2 cooks, 7 servers, and 3 dishwashers?

12C7 * 5C3 * 2C2

The 7 servers can be arranged in 12C7 ways
Once these are selected for each and every one of these combinations there remains 5 workers to allocate beteween dishwashers and cooks
The 3 dishwashers can be arranged in 5C3 ways
This then leaves 2 workers who are automatically cooks 2C2

12C7 = 12! / 7!5! = 12 x11x10 x9 x 8 /(5x4x3x2x1) = 95040/120 = 792
5C3 = 5! / 3!2! = 5x4 /2 = 10
2C2 = 2!/2! = 1

Therefore there are 792 x 10 x 1 = 7920 different arrangements possible

Please note that it makes no difference which order you select the jobs in. If you chose the cooks first any two from 12 = 12C2, then dishwashers any 3 from 10 = 10C3 and then the servers any 7 from 7 = 7C7. Multiply these out
12C2 = 12 x 11 / 2 = 66
10C3 = 10 x 9 x 8 / (3x2x1) = 720 / 6 = 120
7C7 = 7! / 7! = 1
66 x 120 x 1 = 7920

explanation of the maths
12C7
The first server selected can be any of 12
The second server selected can be any of 11 (1 already selected)
The third server can be any of 10 (2 already selected)
The 4th server can be any of 9
The 5th server can be any of 8
The 6th server can be any of 7
The seventh server can be any of 6
Therefore there are 12 x 11 x 10 x 9 x 8 x 7 x 6 = 3991680 permutations of servers that can be selected. Before I go on to explain why this is not the final answer

(this is the same thing as 12! / 5! as if you write out 12x11x10... down to 1 and then divide it by 5x4x3x2x1, the numbers from 5 down to one cancel out leaving just 12 x 11x 10 x 9 x 8 x 7 x 6)

However in all those many permutations of staff there are instances where the same 7 staff are in the same group but were just selected in a different order eg ABCDEFG is exactly the same 7 members as BCDEFGA or GABCDEF or any other permutation In fact there are 7! ie 7x6x5x4x3x2x1 = 5040 permutations for each combination of 7 people. Therefore we must divide the 12! /5! permutations by 7! to reflect this
3991680 / 5040 = 792 (as stated above)
Similar logic applies for the other examples 5C3 and 2C2
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