Answer 1

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

∴ 30240 words can be formed if vowels are always together.