Order, Ranking & Seating

Snapshot

• Order, Ranking & Seating covers four connected skills: ranking in a line ("how many in all", "rank from the other end"), ordering by comparison clues, seating in a row, and seating around a table.
• CUET mostly asks the quick types: total from two ranks, people between two positions, a position after a swap, who sits where in a row of five or six, and a circle of six or eight.
• The idea that cracks it: draw the row or circle first, place the most constrained person, and settle left and right from the way each person faces.
• Marking is +5 / −1. A drawing that takes 40 seconds beats a guess every time.

Part 1 — Ranking in a line: three formulas that do most of the work

Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Prepared for the CUET UG General Test.

A ranking question puts people in a single line (a queue, a class by marks, a row of students) and gives you positions counted from one end. Everything follows from one fact: if a person is $a$-th from the left and $b$-th from the right in a line of $n$ people, then the people ahead of them are $a-1$ and the people behind are $b-1$, so

$$a + b = n + 1 \quad\Longrightarrow\quad n = a + b - 1$$

Three more results come straight out of this and are worth memorising.

"Between" never counts the two persons named. "Ahead of" and "behind" never count the person. These small word differences cost most of the marks lost on ranking questions.

When positions change (two people swap, one steps forward, or one person joins), keep the total fixed unless the question adds or removes people. Work out the position the person now holds, find the matching old position of the other person, and then use $a+b=n+1$.

Part 2 — Ordering by comparison clues

In an ordering question the line is hidden, and clues such as "A is taller than B" must be assembled into a single order.

Write each clue as a chain with the larger on the left: $A > B$, $C > D$. Join chains through a shared name until one chain remains. If two chains share no name, the order between them is not fixed, and a question that asks about it has the answer "cannot be determined". The words to translate are straightforward: "taller", "older", "heavier" and "scores more" mean greater; "shorter", "younger" and "scores less" mean smaller; "the tallest" is the first in the chain; "second from the bottom" is the second-last. Always check that the chain has all the people in it before you answer.

Part 3 — Seating in a row and the left-right rule

The picture shows the same page order P Q R S T in two rows. In the top row everyone faces North, and in the bottom row everyone faces South. In the top row P sits on Q's left. In the bottom row, the same page position makes P sit on Q's right. Nothing about the page has changed: the people turned around, so their left and right changed with them.

Use this checklist for every row:

1. Write the facing direction above the row.
2. Number the seats 1 to $n$ in the order the sitters would count from their own left (for North-facing that is West to East, for South-facing that is East to West).
3. Place the extreme-end clues and the "exactly in the middle" clues first.
4. Treat "immediate" as adjacent, "second to the right" as two seats away, and "between" as strictly inside.
5. Only when the facing is mixed (some North, some South), convert each person's left and right separately.

Part 4 — Seating around a table

Seats on a circle are described by clockwise and anticlockwise. The facing direction decides how left and right map onto them.

For a circle of $n$ equally spaced people, the person opposite seat $k$ sits in seat $k + \tfrac{n}{2}$ (subtracting $n$ if that passes $n$), and opposite only exists when $n$ is even. Two seats $d$ apart on the shorter side have $d-1$ people between them, and the other side has $n - d - 1$. If people face in mixed directions, write I (inward) or O (outward) beside each seat and work out each person's left and right on their own.

Part 5 — Letters and digits in position

CUET also dresses ranking in alphabets and numbers. The English alphabet has 26 letters, so a letter that is $a$-th from the left is $(27-a)$-th from the right, which is just $a+b=n+1$ with $n=26$. Know A=1, E=5, J=10, M=13, N=14, S=19, T=20, Z=26 so that you never count on your fingers. For "arrange the letters of a word alphabetically" or "arrange the digits in ascending order", write the new order once, place it under the old order, and compare position by position.

Part 6 — The full menu of question types

Part 7 — Worked examples

Example 1. Total from two ranks. Ravi is 7th from the left and 9th from the right in a row. How many are in the row?

$n = 7 + 9 - 1 =$ 15.

Example 2. Rank from the other end. A boy is 12th from the top in a class of 40. What is his rank from the bottom?

From the bottom $= 40 + 1 - 12 =$ 29th.

Example 3. Between, same end. A is 15th from the left and B is 20th from the left. How many people are between them?

$20 - 15 - 1 =$ 4.

Example 4. Between, opposite ends. In a row of 40, A is 15th from the left and B is 18th from the right. How many people stand between them?

$40 - (15 + 18) =$ 7. Check by moving B to the left side: $40+1-18=23$ from the left, so between $=23-15-1=7$.

Example 5. After a swap. In a row, P is 10th from the left and Q is 15th from the right. When they exchange places, P becomes 18th from the left. How many are in the row?

P now stands where Q stood, so Q was 18th from the left. Q is 15th from the right, so $n=18+15-1=$ 32.

Example 6. Moving along a queue. In a queue of 30, A is 8th from the front. B stands 11 places behind A. What is B's position from the back?

B is $8+11=19$th from the front. From the back, $30+1-19=$ 12th.

Example 7. Building a chain. Ria scores more than Sam but less than Tia. Uma scores less than Sam but more than Vik. Who scores the second lowest?

Chains: $T > R > S$ and $S > U > V$. Joined through S, the order is $T > R > S > U > V$. The second lowest is Uma.

Example 8. A row facing North. Five persons P, Q, R, S and T sit in a row facing North. T sits at the extreme right end. R sits in the middle. Q sits immediately to the left of R. S does not sit next to T. Who sits at the left end, and who sits between R and T?

Number the seats 1 to 5 from the left (West to East). T is in seat 5, R in seat 3, Q in seat 2. S cannot be in seat 4 (next to T), so S takes seat 1, and P takes seat 4. The order is S Q R P T. The left end is S, and the person between R and T is P.

Example 9. A row facing South. Five persons A, B, C, D and E sit in a row facing South. B sits at the extreme left end. D sits second to the right of B. C sits at the extreme right end. E sits immediately to the left of D. Who sits in the middle, and who sits next to C?

Count in the sitters' own left-to-right order, starting at B. B is first, D is second to the right of B so D is third, and C is fifth. E is immediately left of D, so E is second. A fills seat four. The order is B E D A C. The middle person is D, and the person next to C is A. On the page, because they face South, this order runs from East to West.

Example 10. A circle facing the centre. Six persons A to F sit around a table facing the centre. A sits opposite D. B sits immediately to the left of A. C sits opposite B. F does not sit next to A. Who sits immediately to the right of A?

Number the seats 1 to 6 clockwise and put A in seat 1. Opposite is three seats away, so D is in seat 4. Facing the centre, left is clockwise, so B is in seat 2. C sits opposite B, which is seat 5. Seats 3 and 6 remain for E and F. F cannot be next to A, so F takes seat 3 and E takes seat 6. Right of A means anticlockwise, which is seat 6, so the answer is E.

Example 11. The same circle, facing outward. In Example 10, if everyone faced away from the centre with the seating positions unchanged, who would sit immediately to the right of A?

Facing outward, right is clockwise, which is seat 2. The answer is B.

Example 12. Counting round a circle of eight. Eight persons sit around a table facing the centre. X sits third to the right of Y. How many persons sit between X and Y on the shorter side, and who is opposite X?

Number clockwise with Y in seat 1. Right is anticlockwise, so the first right is seat 8, the second is seat 7 and the third is seat 6. X is in seat 6. The shorter arc runs through seats 7 and 8, so 2 persons sit between them. The person opposite X is in seat $6+4-8=2$, which is the person immediately clockwise of Y.

Example 13. Alphabet positions. Which letter is 9th from the right end of the English alphabet, and which is 5th to the right of L?

The 9th from the right is the $(27-9)=18$th letter, which is R. L is the 12th letter and $12+5=17$, which is Q.

Example 14. Rearranging a word and a number. Arrange the letters of PLANET alphabetically. Which letter is 4th from the right? Then, in the number 3825649, arrange the digits in ascending order. How many digits keep their place?

PLANET sorts to A E L N P T. The 4th from the right is the 3rd from the left, which is L. The number 3825649 sorts to 2345689. Compare 3,8,2,5,6,4,9 with 2,3,4,5,6,8,9 position by position. The 4th digit (5), the 5th (6) and the 7th (9) match, so 3 digits keep their places.

Part 8 — Common traps

Part 9 — Rapid-fire practice with answers

Part 10 — A timed-practice routine

In the exam, a ranking formula question should take 20 seconds. A five-person row puzzle with two or three questions should take about three minutes. A circular arrangement with eight people and mixed facing is the one to leave to the end. Always read the sentence "facing the centre / facing outward / facing North" before you read any clue.

Part 11 — Where this connects

Left and right discipline here is the same as turning in Direction Sense. Family-tree thinking from Blood Relations sometimes appears inside seating puzzles ("A's mother sits to the right of…"). The alphabet-position facts are used again in Coding-Decoding, and the chain-building habit is the same one used in Syllogisms.

One-line revision: use $a+b=n+1$ for totals, exclude the named people in "between", fix the facing before using left and right, treat left as clockwise only when facing the centre, and place the most constrained person first.