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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Jenna Farmer · Accepted Answer

Step 1First 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 2That 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&#039;s.The 9 consonants can be ordered in                     9        !                    3        !        2        !              =    30240   ways since there are 3 N&#039;s and 2T&#039;sSo the total possible number of arrangements satisfying the conditions is       252    ⋅    60    ⋅    30240    =    457228800  .

The letters of the word CONSTANTINOPLE are written on 14

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