How many words can be formed by using the letters from the word “DRIVER” such…

Step-by-Step Solution

To find the number of ways to arrange the letters of the word "DRIVER" such that all vowels are together, we use the "block" method.

1. Identify the letters:

   • The word "DRIVER" has 6 letters: D, R, I, V, E, R.

   • Consonants: D, R, V, R (4 letters)

   • Vowels: I, E (2 letters)

2. Use the "Block" Method:

   • Since the vowels (I, E) must always be together, treat them as a single block: (IE).

   • Now, we consider the consonants plus this single vowel block as distinct units to arrange: {D, R, V, R, (IE)}.

   • This gives us 5 items to arrange.

3. Account for Repeated Letters:

   • Among these 5 items, the letter 'R' appears twice.

   • The number of ways to arrange these 5 items is 5! / 2! = 120 / 2 = 60.

4. Arrange Within the Block:

   • The 2 vowels within the block (I, E) can be arranged among themselves in 2! ways.

   • 2! = 2 * 1 = 2.

5. Calculate Total Arrangements:

   • Total arrangements = (Arrangements of items) * (Arrangements within vowel block)

   • Total = 60 * 2 = 120.