# Find the number of ways of arranging letters of the word MATHEMATICAL. How many of these arrangements have all vowels together? - Mathematics and Statistics

Sum

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!)

Concept: Permutations - Permutations When Repetitions Are Allowed
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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 11th Standard HSC Maharashtra State Board
Chapter 6 Permutations and Combinations
Exercise 6.4 | Q 5 | Page 82