Question

How many five letter words with or without meaning, containing 2 vowels and 3 consonants can be formed using the letters of the word 'OPTICAL'?

• A. 1440
• B. 864
• C. 288
• D. 576

Answer 1

The Correct Option is A

Concept:

• First identify which letters of the given word are vowels and which are consonants.
• Use combinations (nCr) to choose the required number of vowels and consonants separately, since selection order does not matter.
• Once the letters are chosen, arrange all of them using factorial, since arrangement order matters when forming words.

Step 1: Identify vowels and consonants in 'OPTICAL'
Letters: O, P, T, I, C, A, L (7 distinct letters)
Vowels: O, I, A → 3 vowels
Consonants: P, T, C, L → 4 consonants

Step 2: Select 2 vowels out of 3 and 3 consonants out of 4
Ways to choose 2 vowels from 3 = C(3,2) = 3
Ways to choose 3 consonants from 4 = C(4,3) = 4
Total ways to select the 5 letters = 3 x 4 = 12

Step 3: Arrange the chosen 5 letters to form words
Since all 7 letters of OPTICAL are distinct, the 5 selected letters are also all distinct
Number of arrangements of 5 distinct letters = 5! = 120

Step 4: Multiply selections by arrangements
Total words = 12 x 120 = 1440

Final Answer: 1440