1
$\begingroup$

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 ?

$\endgroup$

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.

1
$\begingroup$

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:

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

$\endgroup$
  • $\begingroup$ But, I have a doubt how to know that inclusion exclusion works here ? I have heard only in case of derangements . $\endgroup$ – Jon Garrick Dec 18 '16 at 17:53
  • 1
    $\begingroup$ @Garrick: I'm not sure how I could possibly explain to you why PIE works in this question but not in some other question, when I have no reference to that other question whatsoever. $\endgroup$ – barak manos Dec 18 '16 at 17:55
  • $\begingroup$ Okk !! If I remove the line "Can Inclusion - Exclusion Principle be applied here and Why ?". Then, how to know :) $\endgroup$ – Jon Garrick Dec 18 '16 at 17:56
  • $\begingroup$ @Garrick: I don't know if there's a general method for parsing the text of a question and determining whether it can be solved only by PIE, also by PIE or neither. I just use (my) common sense. $\endgroup$ – barak manos Dec 18 '16 at 17:58
  • $\begingroup$ I got it, that why you applied IE. Thanks !! $\endgroup$ – Jon Garrick Dec 19 '16 at 3:15

Not the answer you're looking for? Browse other questions tagged or ask your own question.