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