Let X = (1,2,3,4,5,6,7,8). Determine the number of permutations
of X which can be expressed as a product of two disjoint cycles, one of length three and the other of length five. Give a brief justification for your solution
Let X = (1,2,3,4,5,6,7,8). Determine the number of permutations
of X which can be expressed as a product of two disjoint cycles, one of length three and the other of length five. Give a brief justification for your solution
There are $\displaystyle \binom{8}{3} = \frac{8!}{3! 5!}$ to divide X into two subsets, one of size 3 and one of size 5.
Given a subset of size 3, how many cycles can we form? There are 3! total orderings of the elements, but each cycle is counted 3 times because there are 3 different places to start listing the cycle. So the total number of cycles is 3! / 3.
Similarly, we can form 5! / 5 different cycles from a subset of size 5.
So all together, there are
$\displaystyle \frac{8!}{3! 5!} \cdot \frac{3!}{3} \cdot \frac{5!}{5}$
permutations of the specified type.