 # The letters of the word CONSTANTINOPLE are written on 14 Frank Day 2022-07-11 Answered
The letters of the word CONSTANTINOPLE are written on 14 cards, one of each card. The cards are shuffled and then arranged in a straight line. How many arrangements are there where no two vowels are next to each other?
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Step 1
First of all just consider the pattern of vowels and consonants.
We are given 5 vowels, which will split the sequence of 14 letters into 6 subsequences, the first before the first vowel, the second between the first and second vowels, etc.
The first and last of these 6 sequences of consonants may be empty, but the middle 4 must have at least one consonant in order to satisfy the condition that no two vowels are adjacent.
That leaves us with 5 consonants to divide among the 6 sequences. The possible clusterings are $\left\{5\right\}$, $\left\{4,1\right\}$, $\left\{3,2\right\}$, $\left\{3,1,1\right\}$, $\left\{2,2,1\right\}$, $\left\{2,1,1,1\right\}$, $\left\{1,1,1,1,1\right\}$.
The number of different ways to allocate the parts of the cluster among the 6 subsequences for each of these clusterings is as follows:
$\left\{5\right\}:6$
$\left\{4,1\right\}:6×5=30$
$\left\{3,2\right\}:6×5=30$
$\left\{3,1,1\right\}:\frac{6×5×4}{2}=60$
$\left\{2,2,1\right\}:\frac{6×5×4}{2}=60$
$\left\{2,1,1,1\right\}:\frac{6×5×4×3}{3!}=60$
$\left\{1,1,1,1,1\right\}:6$
Step 2
That is a total of 252 ways to divide 5 consonants among 6 subsequences.
Next look at the subsequences of vowels and consonants in the arrangements:
The 5 vowels can be ordered in $\frac{5!}{2!}=60$ ways since there are 2O's.
The 9 consonants can be ordered in $\frac{9!}{3!2!}=30240$ ways since there are 3 N's and 2T's
So the total possible number of arrangements satisfying the conditions is $252\cdot 60\cdot 30240=457228800$.

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