Suppose that we want to list A, B, C, D, E, F into a sequence in a way
that A and B are placed on the same side of C. For example, A, D, B, E, F, C and F, E, D, C, B, A
would be both desirable sequences because A, D, B, E, F, C has A and B on the left side of C and
F, E, D, C, B, A has A and B on the right side of C. However, A, E, C, F, B, D would not be a
desirable sequence. Note that A B and C do not have to be adjacent to each other. How many
different sequences can you make?
There are 6 letters: A, B, C, D, E, F.
Let us choose places for 3 letters: A, B, and C. We can do it in "{6 \\choose 3}=\\tfrac{6!}{3!\\cdot 3!}=20" ways.
Other 3 places are for letters D, E, and F.
We can write down letters A, B, and C in 4 ways: A,B,C; B,A,C; C,A,B; C,B,A.
And letters D, E, and F in 3!=6 ways: D,E,F; D,F,E; E,DF; E,F,D; F,D,E; F,E,D.
So, we can make "20\\cdot 4\\cdot6=480" different sequences.
Answer: 480.
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