TITLE: Permutation of a word with atleast one of the alphabet in it
QUESTION [1 upvotes]: How many possible words do I have if I have: k-different Alphabet with
  the length n and atleast one of every element from the alphabet is in
  the word.

My attempt
For example I got an k-long alphabet = {a, b, c} and the length n = 5. I am creating n - k extra variables x and y. x and y can be every element from the alphabet. What I have to calculate first is the permutation of the set {a, b, c, x, y}. Now I have to multiply it by a certain variable so I can also get the permutation of x and y.
My question
What do I have to multiply it with after calculating the permutation first?
I already look into the striling number and the inclusion-exclusion principle. Do I have to use them to get the answer? I dont see any points use it.

REPLY [1 votes]: Hint: For $i=1,\ldots,k$, let $A_i$ be the set of words that do not have the $i^{th}$ letter. The universal set of all words has cardinality $k^n$. Use inclusion-exclusion to find the value you require, which is:
$$\bigcap_{i=1}^{k}A_i^c.$$
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