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

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

Can Inclusion - Exclusion Principle be applied here and Why ?

## closed as off-topic by Namaste, C. Falcon, Shailesh, Adam Hughes, user223391 Dec 19 '16 at 4:20

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Namaste, C. Falcon, Shailesh, Adam Hughes, Community
If this question can be reworded to fit the rules in the help center, please edit the question.

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: $\binom{3}{\color\red1}\cdot\color\red1^5$

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