Maths Olympiad - Permutations 1

Question:

The letters of the words MATHEMATICS are arranged in such a way that the first four letters are all vowels. Find the total no. of distinct arrangements that can be formed this way.


Answer:

There are 4 vowels - AAEI with the letter A appearing twice
There are 7 consonants - MTHMTCS with the letters M and T appearing twice

Using factorials, the no. of distinct combinations of vowels are:
4!/2! = 4 x 3 x 2 x 1 / 2 x 1 = 12 (divide by 2! because the letter A appears twice)

*Note: The formula is the factorial of the total no. of letters divided by the the multiplication of the factorials of the total no. of repeated letters.

Using factorials, the no. of distinct combinations of consonants are:
7!/(2! x 2!) = 7 x 5 x 4 x 3 x 2 x 1 / (2 x 1)(2 x 1) = 210
(divide by 2! twice because the letters M and T appear twice)

Therefore, the total no. of distinct arrangements that can be formed where the first four letters are all vowels = 12 x 210 = 2520