There are 3 men and 3 women to be seated in a row of 10 chairs. In how many diļ¬erent ways can they be seated if one man must be seated at each end of the row?
I began by calculating $_3P_2 = 6$ for the possible combinations for the end seats. The book gives the answer as 10,080, which I obtained by multiplying the first result by $_8P_4 = 1680$. I am unsure, however, about this second part of the problem, and I really only got there after trial-and-error, so I don't really understand why it works.
My understanding of the permutation formula $_nP_k$ is that it is for selecting $k$ objects from $n$ objects when order matters. So, if I were selecting $4$ people to fill $4$ remaining seats, it would make sense to me, but I don't understand where the $4$ empty seats come into play. Shouldn't there be many more possible permutations when accounting for the different positions of the empty chairs?
Best Answer
Ignore "permutation" formulas and just do this directly via rule of product. (It is afterall from the rule of product that we get the permutation formulas in the first place).
Multiplying the number of options for each step together yields the total:
$3\cdot 2\cdot 8\cdot 7\cdot 6\cdot 5 = 10080$