In the word BUCKET, how many pairs of letters have as many letters between them in the word as they have between them in the English alphabet?

Introduction / Context:
This is another letter pair counting question similar to the ones involving CREATIVE and LEMON. The word here is BUCKET, and we must count how many letter pairs satisfy a particular positional property. Such questions help test attention to detail and the ability to apply the same logical rule consistently. They are standard fare in many clerical and banking examinations.

Given Data / Assumptions:
The word given is BUCKET. We must consider all pairs of letters in this word taken from left to right. For each pair, we count the number of letters between them in the word itself and compare that value to the number of letters between them in the usual A to Z alphabet. If the two counts are equal, that pair is counted as one valid pair. The final question asks for the total number of such valid pairs in BUCKET.

Concept / Approach:
As before, we assign alphabetical positions to each letter: A is 1, B is 2 and so on. For any two letters, the number of letters between them in the alphabet is the absolute difference of their positions minus one. Within the word, the letters are at word positions starting from 1, and the number of letters between them is the index difference minus one. We systematically check all pairs to see where these two gap counts agree. A structured approach avoids omissions and double counting.

Step-by-Step Solution:
Write BUCKET with positions: B(1), U(2), C(3), K(4), E(5), T(6). Now write alphabetical positions: B = 2, U = 21, C = 3, K = 11, E = 5, T = 20. Consider the pair C and E at positions 3 and 5 in the word. Between them in BUCKET is one letter, K, so there is 1 letter between them in the word. In the alphabet, between C (3) and E (5) is the letter D, again exactly 1 letter. So C E is a valid pair. If you check other pairs such as B K, B E, U C, U T, C K and so on, the number of letters between them in the word and in the alphabet will not match. Thus, only one pair, namely C E, satisfies the required condition.

Verification / Alternative check:
As a verification method, try listing all possible pairs and writing two small numbers next to each pair: the gap in the word and the gap in the alphabet. For example, for B and C, the word gap is 1 (U between them), but the alphabet gap between B and C is 0. This mismatch rules out that pair. Doing this systematically for every combination shows only C and E have equal gaps. Another cross check is to focus on letters that are close together in the word and see whether their alphabetic spacing is similarly small. Among these, only C and E line up perfectly, confirming the count of one valid pair.

Why Other Options Are Wrong:
Options Two, Three, Four and More than four all claim that there are multiple such pairs. However, detailed analysis reveals only a single valid pair in BUCKET. Any additional pair suggested by those counts will have unequal gaps between word positions and alphabet positions. Since the rule is very strict, even a difference of one letter invalidates a pair. Therefore any answer other than One contradicts the actual pattern in the word.

Common Pitfalls:
One common mistake is to miscount the number of letters between two positions in the word, especially when the pair is near the start or end. Another pitfall is to compare raw alphabetical positions instead of comparing the gap between them. Students sometimes also stop after finding one pair and assume there must be more, without checking. To avoid such issues, keep your method consistent for each pair: compute word gap, compute alphabet gap, and compare. Writing the positions on paper and working slowly at first will greatly improve accuracy with practice.

Final Answer:
There is exactly One such pair of letters in the word BUCKET that satisfies the given condition.