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?

1 Answer




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: