Think of this problem in terms of "slots" that contestants are placed into (first slot, second slot, and so on).
When a decision is made to place a contestant into the first slot, there are 12 possible choices. AFTER this choice has been made, it is then compounded with the choice of the next slot which has 11 people to choose from. This process goes on and on until the last slot is filled (slot twelve) where there is only 1 choice of a contestant.
Based on the the idea of compounding choices, each number of choices for one decision should be multiplied by the number of choices for each following decision. This means that for 12 contestants there are
12*11*10*...*2*3*1 = 12! possible ways to arrange the contents in a given order.
Now lets assume that the order has to alternate going MFMFMFMFMF....
If the selection starts with a Female first, then there are 6 choices. This means then that the following person would have to be Male and there are 6 possible choices for that slot. On the third slot, it must be female but there has already been one female selected so there are only 5 possible choices. This process continues until all 12 slots are filled as shown
slot 1 2 3 4 5 6 7 8 9 10 11 12

gender F M F M F M F M F M F M

# of contestants 6 6 5 5 4 4 3 3 2 2 1 1
As you can see this can be rearranged to give 6*5*4*3*2*1 * 6*5*....*2*1
which is the same as
6! * 6!
Overall, this problem uses both the idea of permutations and the fundamental principle of counting.
4/9/2014

David R.
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