On the island of Mumble, the Mumblian alphabet has only $5$ letters, and every word in the Mumblian language has no more than $3$ letters in it. How many words are possible? (A word can use a letter more than once, but $0$ letters does not count as a word.)
I need to count how many words of different lengths are possible with $5$ letters. For words of length $1$, there are $5$ choices for the letter, so there are $5$ words of length $1$. For words of length $2$, there are $5$ choices for the first letter and $5$ choices for the second letter, so there are $5 \times 5 = 25$ words of length $2$. For words of length $3$, there are $5$ choices for the first letter, $5$ choices for the second letter, and $5$ choices for the third letter, so there are $5 \times 5 \times 5 = 125$ words of length $3$. To get the total number of words, I need to add up the words of different lengths: $5 + 25 + 125 = 155$. 