Say we want to construct a 5 letter word from the English alphabet $E=\{\mathsf a, \mathsf b, \dots,\mathsf z\}$, with repetition not allowed.
Suppose for the moment, that repetition is allowed. In that case, the set of words is simply $E^5$, whose size is $|E^5|=|E|^5=26^5$.
Now in the case where repetition is not allowed, then a word can be constructed in $^{26}\rm P_5 = 26 \times 25 \times 24 \times 23\times 22$ ways, a straightforward calculation.
My question is, is there an easy way to express the corresponding set of tuples we are enumerating, in terms of $E$? All I could think of was the cumbersome
$$\bigcup_{\ell_1\in E}\big(\bigcup_{\ell_2\in E\smallsetminus\{\ell_1\}}\big(\bigcup_{\ell_3\in E\smallsetminus\{\ell_1,\ell_2\}}\big(\bigcup_{\ell_4\in E\smallsetminus\{\ell_1,\ell_2,\ell_3\}}\{\ell_1\}\times\{\ell_2\}\times\{\ell_3\}\times\{\ell_4\}\times(E\smallsetminus\{\ell_1,\ell_2,\ell_3,\ell_4\})\big)\big)\big),$$ which seems needlessly complicated for a set which is so conceptually simple.
Note: I'd like something which makes use of set operations rather than set comprehension like $$\{(\ell_1,\ell_2,\ell_3,\ell_4,\ell_5):\text{$\ell_i\in E$ and $\ell_i\neq \ell_j$ for $i\neq j$}\},$$ which feels a bit like cheating.
The collection of $n$-tuples of a set $X$ without repetitions has the same cardinality as the set of injections from n to $X$, where n is your favorite $n$-element set. So $\mathrm{Inj}(\mathrm n,X)$ seems like a succinct notation, as is $X^\underline n$ in the comments.