Sum

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

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#### Solution

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.

∴ The number of arrangement = ^{7}P_{7} i.e., 7!

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

∴ Required number of arrangements

= 7! × 3!

= 5040 × 6

= 30240

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

Concept: Concept of Permutations - Permutations When Repetitions Are Allowed

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