# How many arrangements can be made with the letters of the word 'MATHEMATIC'? In how many of them will vowels be together?

Sep 7, 2017

Mathematic can be arranged in 453,600 different ways if it is ten letters and only use each letter once. Assuming all vowels will be together 15,120 arrangements.

#### Explanation:

Requires work with permutations and factorials. Factorial is written as '!' . Factorial is the multiplication of all it lower terms.
Eg 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Mathematic has ten letters and as such can be arranged 10! ways. As it has repeating letters you divide by this repetitions.

As such it becomes (10!)/(2!*2!*2!) . This equation equals 453 600

For the second part it should be treated as two parts. All the vowels are grouped together so there are effectively only 7 letters left.

This you would write as (7!)/(2!*2!) (Lose a 2! as the repeating vowel is not counted.) Then for the vowel, it is (4!)/(2!).
You now multiply these together to get your answer of 15120