Question

The number of words (with or without meaning) that can be formed from all the letters of the word "LETTER" in which vowels never come together is _ _ _ _ _ .

Hint:

For "vowels never together", always: 1) Arrange consonants 2) Place vowels in gaps This avoids double counting.

Answer 1

Correct Answer: 120

Concept: Use gap method for "vowels never together".

Step 1: Identify letters Word: LETTER \[ L, E, T, T, E, R \] Vowels: \(E, E\) Consonants: \(L, T, T, R\)

Step 2: Arrange consonants \[ \text{Number of ways} = \frac{4!}{2!} = \frac{24}{2} = 12 \]

Step 3: Create gaps \[ \_ \; C \; \_ \; C \; \_ \; C \; \_ \; C \; \_ \] Total gaps = 5

Step 4: Place vowels Choose 2 gaps out of 5: \[ \binom{5}{2} = 10 \] Arrange vowels \(E, E\): \[ \frac{2!}{2!} = 1 \]

Step 5: Total arrangements \[ \text{Total} = 12 \times 10 \times 1 = 120 \] Final: 120