15.3 Ranking, People Between & Position Interchanges

Order and Ranking Mechanics

Order-ranking problems assess positions in a linear hierarchy (seniority lists, arrival queues, platform track allocations).

1. Fundamental Positional Formulas

For a single linear row of items or persons:

$$\text{Total Persons } (N) = \text{Rank from Left } (L) + \text{Rank from Right } (R) - 1$$

• Why subtract 1? The focal individual is counted twice—once in the leftward tally and once in the rightward tally.
• Derived Formula for Rank from Opposite End:

$$R = N - L + 1, \quad L = N - R + 1$$

2. Number of Persons Between Two Individuals

Given two individuals $A$ (rank $L_A$ from left) and $B$ (rank $R_B$ from right) in a cohort of size $N$:

Convert B to left position $L_B=N-R_B+1$. For distinct A and B:

$$\text{Persons between}=|L_A-L_B|-1$$

Adjacent positions give zero. Equal positions cannot represent two distinct people in a strictly ordered row. This formula handles either order and avoids missing the adjacency boundary case.

3. Position Interchange Problems

When two individuals $A$ and $B$ swap positions, individual $A$ moves into individual $B$'s original seat:

$$\text{Total Cohort } (N) = \text{New Rank of } A + \text{Original Rank of } B \text{ from other end} - 1$$ $$\text{Shift Distance } \Delta = |\text{New Rank of } A - \text{Old Rank of } A|$$

Use the original seat of A and convert its rank to the requested end after swapping.

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Worked Example 2: Position Interchange in Seniority Roster

Problem: In a divisional seniority list of Section Controllers arranged in a single row, Manoj is 17th from the left end, and Rajesh is 22nd from the right end. When they swap places, Manoj becomes 29th from the left end. Find: (a) the total number of controllers in the roster, and (b) Rajesh's new position from the right end.

• Step 1: Identify Seat Invariance: After swapping, Manoj occupies Rajesh's former seat. Manoj is now 29th from the left end, while that exact seat was originally known to be 22nd from the right end.
• Step 2: Compute Total Controllers ($N$):

$$N = \text{Rank}_{\text{Left}} + \text{Rank}_{\text{Right}} - 1 = 29 + 22 - 1 = 50$$

• Step 3: Compute Rajesh's New Rank from Right: Rajesh now occupies Manoj's original seat, which was 17th from the left end.

$$\text{Rank}_{\text{Right}} = N - \text{Rank}_{\text{Left}} + 1 = 50 - 17 + 1 = 34$$

Alternatively, Manoj shifted right by $29 - 17 = 12$ places. Rajesh must shift left by the identical $12$ places: $22 + 12 = 34$th from the right end.

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Convert every rank to one end

The most reliable way to count people between two positions is to express both positions from the same end. If A is L places from the left and B is R places from the right in a row of N distinct people, B's left position is N minus R plus one. For different people, the number between them is the absolute difference of their left positions minus one. This covers both orders without memorizing overlapping cases.

For example, in a row of 40, A is 12th from the left and B is 9th from the right. B is 32nd from the left, so 19 people are between positions 12 and 32. In the same row, if B were 35th from the right, B would be 6th from the left and five people would be between them. Reversing their order does not make the count negative.

If both descriptions identify the same person, the positions coincide and the distinct-person formula is inapplicable. Check whether the problem guarantees two different individuals. Likewise, a ranking with ties is not automatically a row of distinct consecutive positions; use the tie convention actually specified.

During a swap, seats stay fixed while occupants exchange. Draw the two original seat numbers, then move the names. A person's rank from a given end can increase or decrease depending on the direction of movement. The two people move equal distances in opposite directions; there is no universal rule that both new ranks equal old rank plus the shift.

Do not infer total group size from one person's rank at only one end. A person 17th from the left could be in many different row sizes. A left/right pair for the same fixed seat, or another independent constraint, is needed to determine N uniquely.

Test whether the rank information is possible

In a row of 25 distinct people, a person 10th from the left is 16th from the right, because 25 - 10 + 1 = 16. A claim that the same person is also 18th from the right contradicts the fixed row size: 10 + 18 - 1 would require 27 people. Do not average conflicting ranks.

If two different people are each described as 10th from the left in the same strictly ordered row, the clues are inconsistent unless a tie or shared-position convention is supplied. Ranking arithmetic relies on the meaning of positions as well as numerical operations. Identify impossible data rather than forcing an answer from whichever formula uses the most numbers.