How many pairs of digits can be formed in the number 73915243, which have the…

Concept: Write the digits 0-9 out as the natural number series. For any two DISTINCT digit-values a and b, the number of digits lying between them in that series is |a - b| - 1 (equal digit-values have no series-gap to compare, so such pairs are set aside). A pair of digits inside a given number is said to "match the series" when the number of digits lying between their positions in the number equals this value-gap. Because the series is a straight line, reading it forward (0 to 9) or backward (9 to 0) gives the identical gap, so checking "both directions" never changes a result - every position-pair needs checking only once.

Applying this to 73915243: first lay out each digit against its position.

Now compare the positional gap against the value gap for every pair of positions with distinct digit-values (the repeated digit 3 at positions 2 and 8 is set aside, since equal values have no series-gap). Five pairs come out equal:

1. 7 (position 1) and 9 (position 3): one digit lies between their positions (the 9's neighbour, digit 3); in the series 7 and 9 also have exactly one digit between them (the 8). The gaps match.

2. 7 (position 1) and 2 (position 6): four digits lie between their positions (3, 9, 1, 5); in the series 7 and 2 also have exactly four digits between them (3, 4, 5, 6). The gaps match.

3. 3 (position 2) and 1 (position 4): one digit lies between their positions (9); in the series 3 and 1 also have exactly one digit between them (2). The gaps match.

4. 1 (position 4) and 4 (position 7): two digits lie between their positions (5, 2); in the series 1 and 4 also have exactly two digits between them (2, 3). The gaps match.

5. 4 (position 7) and 3 (position 8): they sit next to each other, so zero digits lie between their positions; in the series 4 and 3 are also next to each other, giving zero digits between them. The gaps match.

That is five position-pairs whose gap matches the series gap - more than three, which is the option this settles on.

Cross-check: Running through every remaining combination of the eight positions (other than the repeated-digit pair) confirms no sixth match turns up, and since the series gives an identical gap whichever direction it is read, re-checking it backward changes none of the five results - the count holds at five.