Words containing all three letters

Text

How many $5$-letter “words” formed from the alphabet $\{A, B, C\}$ contain each of the three letters at least once?

Solution

We use the inclusion-exclusion principle. The total number of words is $3^{5}=243$. We subtract those missing at least one letter.

• Words using at most $2$ fixed letters (one is missing): $2^{5}=32$ each, and the missing letter can be chosen in $\binom{3}{1}=3$ ways $\Rightarrow 3\cdot 32 = 96$.
• But this removed twice the words with a single letter; there are $1^{5}=1$ each, for $\binom{3}{2}=3$ pairs $\Rightarrow$ we add them back: $+3$.

Hence the “complete” words number $N = 3^{5} - 3\cdot 2^{5} + 3\cdot 1^{5} = 243 - 96 + 3 = \boxed{150}.$