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# How many different words using all letters of ZOONOOZ can be formed if there are no O's together?

## How could you show this mathematically?

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Jun 19, 2018

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#### Explanation:

In the word ZOONOOZ, we have 7 letters with 4 Os and 2 Zs. We don't want any of the Os together.

Let's first notice that the only way to arrange the 4 Os are in places 1, 3, 5, 7. There's only 1 way to arrange them.

We then have the 2 Zs and the 1 N to arrange across 3 spots (places 2, 4, 6). If the three letters were all different, there'd be 3! = 6 ways to arrange them. However, because we have duplicate Zs, we need to divide by the number of ways they can be arranged within themselves, which is 2! = 2. This gives:

$\frac{6}{2} = 3$

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