In the word TROUBLED, how many pairs of letters are there such that the number of letters between them in the word is the same as the number of letters between them in the English alphabet?

Introduction / Context:
This question tests careful counting and understanding of alphabetical positions. We are asked to find pairs of letters in the word TROUBLED where the gap between the letters in the word matches the gap between the same letters in the English alphabet, measured in terms of letters in between.

Given Data / Assumptions:

• Word: TROUBLED.
• Letters in order: T, R, O, U, B, L, E, D.
• Alphabetical positions are based on A = 1, B = 2, and so on up to Z = 26.
• For a pair of letters, we compare the number of letters between them in the word and in the alphabet.

Concept / Approach:
For any pair of letters at positions i and j in the word (i less than j), the number of letters between them in the word is (j minus i minus 1). In the alphabet, if their positions are p1 and p2, the number of letters between them is the absolute value of (p1 minus p2) minus 1. We need to find pairs where these two counts are equal.

Step-by-Step Solution:
Step 1: Write the word with indices: T(1), R(2), O(3), U(4), B(5), L(6), E(7), D(8). Step 2: For each pair of positions, compute the word gap j - i - 1. Step 3: Map each letter to its alphabet position (for example, A = 1, B = 2, ..., Z = 26). For O and L, O is 15, L is 12 and so on. Step 4: For each pair, compute the alphabet gap abs(p1 - p2) - 1. Step 5: Check equality of the word gap and alphabet gap. Step 6: The pairs that satisfy this condition are O and L, and E and D.

Verification / Alternative check:
You can verify individual pairs explicitly. For example, O is at position 3 in the word and L is at position 6, so there are 6 - 3 - 1 = 2 letters between them in TROUBLED. In the alphabet, O is letter 15 and L is letter 12, so there are abs(15 - 12) - 1 = 2 letters between them as well. A similar verification works for E and D.

Why Other Options Are Wrong:
Options 3, 4 and 5 assume more matching pairs than exist. A detailed check shows only two valid pairs satisfy the equal gap condition, so those larger counts are overestimates.

Common Pitfalls:
Many students miscount letters between positions or forget to subtract one when converting positions into counts of letters in between. Another common issue is to miscalculate alphabetical positions for letters, especially later in the alphabet.

Final Answer:
The number of such pairs of letters in TROUBLED is 2.