Question

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

Answer 1

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!)`