Given the set {a,b,c,d,e,f}, how many 4 letter strings contain the word ace?
This is a subset arrangement. My reasoning is that there are four positions into which to place 6 letters. Total permutations would be 6x5x4x3 = 360. Then with the condition that the letters a,c,e be together in that specific order means that they would now occupy three fixed positions with one position remaining. That last position can be filled with 3 letters b,d,f. Also a,c,e could be justified right or justified left in the four positions. Does this mean that the answer to the question is 2!x3! Or 2! x 3?
Hint: The number of possible arrangements is small. So that you can easily list all of them. The pattern is as your correctly stated
\begin{align*} Xace\qquad\text{or}\qquad aceX \end{align*}
with $X\in\{b,d,f\}$. Do you see the correct answer?