There are eighteen guard posts. How many ways can 25 not indistinguishable (That is, they are distinguishable.)guards be distributed to the guard posts, so that no post is empty?

Question

Try - (25 choose 18) ??

Answer 1

Think of it this way:

Every time all the 18 posts are filled, there will be a set of 7 guards left out.
So the question becomes, how many set of 7 from 25 can we find, which is
C(25,7) or C(25,8) , both yield the same answer.

BTW, your clarification in brackets is a contradiction
you probably meant to say
((That is, they are NOT distinguishable.)