In how many distinguishable rearrangements of the letters ABCCDEEF does the A precede both C's, the F appears between the 2 C's, and the D appears after the F?
Source: Purple Comet Math Meet 2011: High School Level
The letters A, C, C, D, F must be arranged in the order as in either of the sequences ACFCD and ACFDC.
For each of these sequences, we then create 3 positions to place the remaining 3 letters B, E, E.
The number of ways to create these positions in each sequence is the same as the number of
permutations of ***XXXXX, which is (5+3)C3 = 8*7*6/3*2 = 56. Lastly, the letters B, E, E
can be put into each set of 3 positions in 3!/2! = 3 ways. In all, the number of distinguishable
rearrangements of the letters ABCCDEEF is 2*56*3 = 336.
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