How many such pairs of digits are there in the number 421579368, each of which has as many digits between them in the number as they have between them when the two digits are arranged in ascending order?

Introduction / Context:
This time-sequence reasoning question asks you to count specific pairs of digits in a given number. The condition links the physical distance between the digits in the given number with their distance in the natural ascending order of digits. Such problems test precision in counting and understanding of positional relationships.

Given Data / Assumptions:

- The number is 421579368.- We must consider all possible pairs of digits from this number.- For each pair, we compare the number of digits between them in the given number with the number of digits between them in ascending numerical order.

Concept / Approach:
For a pair of digits (x, y), first arrange them in ascending order as (smaller, larger). Count how many digits lie between them in the standard number order 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. That count equals larger minus smaller minus 1. Then, in the given number, count how many digits lie between the two positions where these digits occur. If these two counts match, the pair satisfies the condition.

Step-by-Step Solution:
Step 1: Write the digits with positions: 4(1), 2(2), 1(3), 5(4), 7(5), 9(6), 3(7), 6(8), 8(9).Step 2: Systematically examine pairs and compute two quantities: (a) position difference minus 1 in the given number and (b) value difference minus 1 when the digits are ordered ascending.Step 3: For the pair 4 at position 1 and 9 at position 6, digits between in the number are 6 - 1 - 1 = 4. In ascending order from 4 to 9, we have 5, 6, 7, 8, which is also 4 digits. This pair qualifies.Step 4: For the pair 2 at position 2 and 1 at position 3, there are 3 - 2 - 1 = 0 digits between them in the number. In ascending order, the pair is 1, 2, with no digits between them, so this pair also qualifies.Step 5: For the pair 1 at position 3 and 6 at position 8, there are 8 - 3 - 1 = 4 digits between them in the number. In ascending order from 1 to 6, we have 2, 3, 4, 5, again 4 digits, so this pair qualifies as well.Step 6: Checking all other pairs shows that only these three pairs satisfy the condition.

Verification / Alternative check:
Once the three valid pairs are identified, recheck the counts: (4, 9) have four digits between them in the number and four digits between them in ascending order. (1, 2) have zero digits in both cases, and (1, 6) have four digits in both cases. This confirms that exactly three pairs meet the requirement.

Why Other Options Are Wrong:
Option None is wrong because we have clearly found multiple pairs that satisfy the condition. Options One and Two underestimate the count by ignoring some valid pairs. Only option Three matches the detailed pair-wise verification across the entire number.

Common Pitfalls:
Candidates may miscount the number of digits between positions or forget to reorder the digits in ascending order before checking the value difference. Another frequent error is to stop searching after finding one or two pairs, assuming no others exist. A systematic check of all pairs is essential in this type of question.

Final Answer:
The number of such qualifying pairs of digits in 421579368 is three.