Sum

Determine the number of arrangements of letters of the word ALGORITHM if no two vowels are together

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#### Solution

In the word 'ALGORITHM' the number of letters is n = 9.

No two vowels are together:

Consider one of the possible arrangements of V (vowels) and C (consonants)

VCVCVCVCVCVCV

Here, there are 7 positions for vowels and 6 positions for consonants.

∴ 3 vowels can be arranged in ^{7}P_{3} ways and 6 consonants can be arranged in ^{6}P_{6} ways.

Hence, the total number of words in which no two vowels are together

= ^{7}P_{3} × ^{6}P_{6}

= `(7!)/(4!) xx 6!`

= 7 × 6 × 5 × (6 × 5 × 4 × 3 × 2 × 1)

= 210 × 720

= 151200

Concept: Concept of Permutations - Permutations When Repetitions Are Allowed

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