Question

Determine the number of arrangements of letters of the word ALGORITHM if vowels are always together.

Answer 1

A word is to be formed using the letters of the word ALGORITHM.
There are 9 letters in the word ALGORITHM.
When vowels are always together:
There are 3 vowels in the word ALGORITHM. (i.e., A, I, O)
Let us consider these 3 vowels as one unit.
This unit with 6 other letters is to be arranged.
∴ It becomes an arrangement of 7 things which can be done in ⁷P₇ i.e., 7! ways and
3 vowels can be arranged among themselves in ³P₃ i.e., 3! ways.
∴ Total number of ways in which the word can be formed= 7! × 3! = 30240
∴ 30240 words can be formed if vowels are always together.