Question

Five letter words, having distinct letters, are to be constructed using the letters of the word 'EQUATION' so that each word contains exactly three vowels and two consonants. How many of them have all the vowels together?

• A. 3600
• B. 1800
• C. 1080
• D. 900

Answer 1

Answer: (C)

1080

Answer 2

Here's how to solve this problem:

1. Select the letters:

   • Choose 3 vowels out of 5: $\binom{5}{3} = 10$ ways.
   • Choose 2 consonants out of 3: $\binom{3}{2} = 3$ ways.

2. Arrange the letters with vowels together:

   • Treat the 3 vowels as a single block. Now we have this block and the 2 consonants, giving us 3 units to arrange.
   • Arrange these 3 units: $3! = 6$ ways.
   • Arrange the vowels within their block: $3! = 6$ ways.

3. Calculate the total number of words:

   $$10 \times 3 \times 6 \times 6 = 1080$$