Hi, Fahim,
I note that "the number of permutations of n things" is legitimate terminology, is complete and explicit, and requires no further qualification. This is in fact the basic use, in mathematics today, of the term "permutations".
It is indeed numerically and logically equivalent to "the number of permutations of n things taken n at a time", a special case of another broader meaning of the term "permutation, which you seem to believe is the only legitimate meaning.
It is equivalent to, in your notation, "n things taken r at a time for r = n"
The formal name for such a situation (not often heard) is a "partial permutation", which avoids confusion with "permutation(s)", whose meaning I discussed at the outset.
I realize that when I was first taught about permutations and combinations (ca. 1952, I suspect), the term "permutations" was taught with the meaning you ascribe to it (what is today called, formally, "partial permutation"). That may well have been true for you (not to suggest that we are contemporaries).
The Wikipedia article on this overall topic says, condescendingly, that this meaning (of "permutations") is "sometimes used in elementary combinatorics texts".
Best regards,
Doug
