Question

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

Answer 1

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