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## Homework Statement

a) How many arrangements of the letters in COMBINATORICS have no consecutive vowels?

b) In how many of the arrangements in part (a) do the vowels appear in alphabetical order?

## Homework Equations

C(n,k) P(n,k)

## The Attempt at a Solution

a) First I divided up the consonants and the vowels. My consonants are 2 C's, M, B, N, T, R, and S. My vowels are 2 O's, 2 I's and one A. Now I find the total number of ways to arrange the consonants = 8! Now I have to arrange my vowels such that there are no consecutive vowels. In the diagram below, the K = consonants, and the v's = vowels.

vKvKvKvKvKvKvKvKv

Since there are nine places to place vowels in order to avoid having consecutive vowels, there are C(9,5).

So the solution I am arriving at = 8! * C(9,5) total combinations.

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