Question

All possible words (with or without meaning) that contain the word 'GENTLE' are formed using all the letters of the word 'INTELLIGENCE'. Then the number of words in which the word 'GENTLE' appears among the first nine positions only is

• A. 1440
• B. 5040
• C. 2520
• D. 720

Hint:

In permutation problems with a "contiguous block" constraint, always treat the block of letters as a single item. First, calculate the number of ways to place this block and the other items. Then, multiply by the number of ways to arrange the letters within the block itself (if applicable, here it's just 1 way since 'GENTLE' is fixed).

Answer 1

The Correct Option is A

First, let's analyze the letters in the word 'INTELLIGENCE'.
The letters are: I(2), N(2), T(1), E(3), L(2), G(1), C(1). Total of 12 letters.
The word to be contained is 'GENTLE'. The letters used are G, E, N, T, L, E.
The letters remaining after forming 'GENTLE' are: I(2), N(1), E(1), L(1), C(1). There are 6 remaining letters.
The question requires that the word 'GENTLE' appears as a contiguous block. Let's treat 'GENTLE' as a single unit or block, let's call it 'B'.
The condition "appears among the first nine positions only" means the entire block 'B' must be contained within positions 1 to 9.
The block 'B' has a length of 6 letters. The total word has 12 positions.
If the block 'B' starts at position 1, it occupies positions 1-6. This is within the first nine.
If 'B' starts at position 2, it occupies 2-7. This is within the first nine.
If 'B' starts at position 3, it occupies 3-8. This is within the first nine.
If 'B' starts at position 4, it occupies 4-9. This is within the first nine.
If 'B' starts at position 5, it occupies 5-10. This is NOT fully within the first nine positions.
So, the block 'B' can start at positions 1, 2, 3, or 4. This gives 4 possible starting positions for the block 'GENTLE'.
For each of these 4 placements of the block 'B', we must arrange the 6 remaining letters in the 6 remaining empty slots.
The remaining letters are {I, I, N, E, L, C}.
The number of ways to arrange these 6 letters, with the letter 'I' repeated twice, is given by \( \frac{6!}{2!} \).
Number of arrangements = \( \frac{720}{2} = 360 \).
The total number of words is the product of the number of possible positions for the block and the number of arrangements of the remaining letters.
Total words = (Number of starting positions) \(\times\) (Arrangements of remaining letters) = \(4 \times 360 = 1440\).