Determine the number of arrangements of letters of the word ALGORITHM if vowels are always together
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.
∴ The number of arrangement = 7P7 i.e., 7!
3 vowels can be arranged among themselves in 3P3 i.e., 3! ways.
∴ Required number of arrangements
= 7! × 3!
= 5040 × 6
∴ 30240 words can be formed if vowels are always together.