"Given a set $\{1,\ 2,\ 3,\ 4\}$, how many sequences with a length of $4$ with entries from this set have exactly one entry equal to $1$?"
Here is my work so far:
$X = \left\{\text{sequences with length 4 from}\ \{1, 2, 3, 4\}\ \text{with exactly one entry equal to $1$}\right\}$ $Y = \left\{\text{permutations of length 4 from}\ \{1, 2, 3, 4\}\right\}$
By definition, $|Y| = 4\times 4\times 4\times4 = 256$.
Where do I go from here? Should I find $|X|$ as well? I think (by intuition, I haven't checked) that I need to find $|X \cap Y|$ in order to solve this problem. Is this true?