Words with at least one E

Text

Using a $21$-letter alphabet, how many $5$-letter “words” (meaningful or not) contain the letter E at least once?

Solution

It is easier to count by complement. The total number of $5$-letter words is $21^5 = 4\,084\,101.$ The words that do not contain E use only the other $20$ letters: $20^5 = 3\,200\,000.$ Hence the words with at least one E number $N = 21^5 - 20^5 = 4\,084\,101 - 3\,200\,000 = \boxed{884\,101}.$