How many $5$ letter words that use letters from the $3$ letter set ${a,b,c}$ in which each letter occurs at least once?

Question

How many $5$ letter words that use letters from the $3$ letter set ${a,b,c}$ in which each letter occurs at least once ?

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Can Inclusion - Exclusion Principle be applied here and Why ?

Answer 1

You can use Inclusion/Exclusion Principle as follows:

• Include the number of words with at most $\color\red3$ types of letters: $\binom{3}{\color\red3}\cdot\color\red3^5$
• Exclude the number of words with at most $\color\red2$ types of letters: $\binom{3}{\color\red2}\cdot\color\red2^5$
• Include the number of words with at most $\color\red1$ types of letters: $\binom{3}{\color\red1}\cdot\color\red1^5$

The answer is therefore:

$$\sum\limits_{n=0}^{2}(-1)^n\cdot\binom{3}{3-n}\cdot(3-n)^5$$