## Permutations and Combinations

### 2.3 Aside: Making Sense of 0!

Lesson materials located below the video overview.

 Consider the “word” CHEESIESTESSNESS, the quality of being the cheesiest of cheeses. Do you see that there are$$\dfrac{16!}{5!6!}$$ ways to arrange its letters?

Exercise 13: Actually evaluate this number.

This may seem strange, but it is actually better to write this answer as:

$$\dfrac{16!}{1!1!5!6!1!1!1!}$$

$$1!$$ for the one letter C

$$1!$$ for the one letter H

$$5!$$ for the five letters E

$$6!$$ for the six letters S

$$1!$$ for the one letter I

$$1!$$ for the one letter T

$$1!$$ for the one letter N

This offers a self check: The numbers appearing on the bottom should sum to the number appearing on the top.

Question:  Back to $$1!$$. Does the definition of factorial actually make sense for $$1!$$? In this context, we want $$1!$$ to equal one so that the value of the denominator is still $$5!6!$$. So maybe the mathematics here is suggesting that we should set $$1!=1$$.

 Exercise 14: In how many ways can you rearrange the letters of your full name?

Carrying on with CHEESIESTESSNESS …

We’ve written the formula for the number of ways to rearrange its letters as:

$$\dfrac{16!}{1!1!5!6!1!1!1!}$$

with one term in the denominator for the count of each letter in the word.

Hmmm. As there are no Ps in this word, should we a actually write this formula as: $$\dfrac{16!}{1!1!5!6!1!1!1!0!}$$ ?

There are also no Qs and no Ms: $$\dfrac{16!}{1!1!5!6!1!1!1!0!0!0!}$$ .

The quantity $$0!$$ doesn’t actually make sense, but what value might we as a society assign to it so that the above formulas still make sense and are correct?

 Exercise 15: Evaluate the following expressions:a) $$\dfrac{800!}{799!}$$   b) $$\dfrac{15!}{13!2!}$$    c) $$\dfrac{87!}{89!}$$  d) $$\dfrac{0!}{0!}$$

All the answers appear in the COMPANION GUIDE to this Permutations and Combinations course.

 Exercise 16: Make the following expressions look significantly simpler:a) $$\dfrac{N!}{N!}$$  b) $$\dfrac{N!}{(N-1)!}$$  c) $$\dfrac{n!}{(n-2)!}$$     d) $$\dfrac{1}{k+1} \cdot \dfrac{(k+2)!}{k!}$$     e) $$\dfrac{n!(n-2)!}{\left((n-1)!\right)^{2}}$$

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