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Sum

Determine the number of arrangements of letters of the word ALGORITHM if vowels are always together.

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#### Solution

A word is to be formed using the letters of the word ALGORITHM.

There are 9 letters in the word ALGORITHM.**When vowels are always together:**

There are 3 vowels in the word ALGORITHM. (i.e., A, I, O)

Let us consider these 3 vowels as one unit.

This unit with 6 other letters is to be arranged.

∴ It becomes an arrangement of 7 things which can be done in ^{7}P_{7} i.e., 7! ways and

3 vowels can be arranged among themselves in ^{3}P_{3} i.e., 3! ways.

∴ Total number of ways in which the word can be formed= 7! × 3! = 30240

∴ 30240 words can be formed if vowels are always together.

Concept: Permutations - Permutations When Repetitions Are Allowed

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