Order & Ranking

Counting between two people

Four shapes cover every 'how many between' question. Know which end each rank comes from — that decides the formula.

Shape 1 — both ranks from the same end. A is a-th and B is b-th from the left:

$\text{between} = |a - b| - 1$

The gap shrinks by one because 'between' excludes both people. Example: 9th and 26th from the top → 16 between.

Shape 2 — ranks from opposite ends. Convert one rank to the shared end first, or use the total. In a row of T with P a-th from the left and Q b-th from the right:

$\text{between} = T - a - b$

Example: row of 45, X 14th from the left, Y 18th from the right → $45 - 14 - 18 = 13$. If the total is not given, build it with $a + b - 1$ from any person who has both ranks.

Shape 3 — the middle person. Both from the same end a and b: the midpoint rank is $(a + b) / 2$ from that end. 9th and 21st from the left → the middle person is 15th from the left. If a + b is odd, the exact middle is a half-position — exams phrase those as 'immediately between', and the two neighbours are $(a+b-1)/2$ and $(a+b+1)/2$.

Shape 4 — the overlap deduction. The classic: 'Ram is 17th from the left, Shyam 9th from the right; after interchanging, Ram is 22nd from the left.' Ram takes Shyam's seat, keeping Shyam's from-right rank (9): total $= 22 + 9 - 1 = 30$. Then Shyam's old seat was $30 - 9 + 1 = 22$nd from the left, and the original gap is $22 - 17 - 1 = 4$. Overlap questions always split into: build the total first, then run a plain Shape 1 or 2 count.

Sanity checks. Between-counts can never be negative — a negative answer means the two ranks described the same person or the ranks were from ends you did not account for. And between + 2 ≤ total: the two people themselves exist too.

Quick revision

• Same end: $|a-b| - 1$. Opposite ends (total T): $T - a - b$.
• Middle person (same end): $(a+b)/2$ from that end.
• Overlap puzzles: recover T first ($a' + b' - 1$), then count between.
• Convert every rank to the same end before subtracting; 'between' excludes both people.
• Negative gap = misread ends; stop and re-convert.

A worked flow for exam speed. Read the question once, labelling each rank with its end (L/R or T/B) and circling the total if present. Then one line of arithmetic:

• same end → subtract, drop 2 (|a−b| − 1);
• opposite ends → subtract from the total;
• no total → build it from any both-ends person (a + b − 1). Under exam pressure the labelling step is what prevents the classic error of subtracting ranks that came from different ends.

Between-counts with a twist. Questions sometimes ask 'how many people do NOT lie between them' (= total − between − 2) or 'what fraction of the row lies between'. Both are one extra subtraction on the between-count — compute between first, always, then answer the actual question.