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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Answers (1)

Jenna Farmer
Answered 2022-07-12 Author has 17 answers
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 { 5 } , { 4 , 1 } , { 3 , 2 } , { 3 , 1 , 1 } , { 2 , 2 , 1 } , { 2 , 1 , 1 , 1 } , { 1 , 1 , 1 , 1 , 1 } .
The number of different ways to allocate the parts of the cluster among the 6 subsequences for each of these clusterings is as follows:
{ 5 } : 6
{ 4 , 1 } : 6 × 5 = 30
{ 3 , 2 } : 6 × 5 = 30
{ 3 , 1 , 1 } : 6 × 5 × 4 2 = 60
{ 2 , 2 , 1 } : 6 × 5 × 4 2 = 60
{ 2 , 1 , 1 , 1 } : 6 × 5 × 4 × 3 3 ! = 60
{ 1 , 1 , 1 , 1 , 1 } : 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 5 ! 2 ! = 60 ways since there are 2O's.
The 9 consonants can be ordered in 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 60 30240 = 457228800 .

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