Not nearly so many words (without repetition)

For a 1-letter word there are 5 options.

For a 2-letter word there are 5 options for the first letter but then only 4 for the second. So there are 5 $\times$ 4 = 20 2-letter words.

For a 3-letter word there are 5 options for the first letter, 4 for the second letter and 3 for the third letter. So there are 5 $\times$ 4 $\times$ 3 = 60 3-letter words.

Working on similar lines for 4- and 5-letter words there are 120 for each.

So we have 5 + 20 + 60 + 120 + 120 = 325 words altogether.

[For those "in the know", this $^5\text{P}_1+\text{ }^5\text{P}_2+\text{ }^5\text{P}_3+\text{ }^5\text{P}_4+\text{ }^5\text{P}_5$ with $^n\text{P}_r = \frac{n!}{(n-r)!}$ .]