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Sum

Find the number of ways of arranging letters of the word MATHEMATICAL. How many of these arrangements have all vowels together?

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#### Solution

There are 12 letters in the word MATHEMATICAL in which ‘M’ repeats 2 times, ‘A’ repeats 3 times and ‘T’ repeats 2 times.

∴ Total number of arrangements = `(12!)/(2!3!2!)`

When all the vowels i.e., ‘A’, ‘A’, ‘A’, ‘E’, ‘I’ are to be kept together Number of arrangements of these vowels = `(5!)/(3!)` ways.

Let us consider these vowels together as one unit.

This unit is to be arranged with 7 other letters in which ‘M’ and ‘T’ repeated 2 times each.

∴ Number of arrangements = `(8!)/(2!2!)`

∴ Total number of arrangements = `(8!xx5!)/(2!2!3!)`

Concept: Permutations - Permutations When Repetitions Are Allowed

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