Question

The inhabitants of the island of Jumble use the standard Kobish alphabet (20 letters, A through T). Each word in their language is 4 letters or less, and for some reason, they insist that all words contain the letter A at least once. How many words are possible?

Answer 1

consisting of 1 letter ---- 1 , namely the A
consisting of 2 letters ::
without restrictions= 20x20 = 400
without the A = 19x19 = 361
at least 1 A ----> 400-361 = 39

consisting of 3 letters, no restricion = 20^3 = 8000
without any A's = 19^3 = 6859
at least one A = 8000-6859 = ....

do the 4 letter case the same way, add up the 4 cases.