PYQ Vault

Playbook

Permutations and Combinations

40 q - 1.00/paper - 40% HARD. The least mechanical chapter on the paper: no formula rescues a misread constraint. Five pages in notes order; Circular Arrangements is 6 q at 83% HARD, the densest corner in the chapter. One question a paper, and the one most likely to eat five minutes.

Questions in the bank
40
q/paper (2024–25 shifts)
1.00
Tagged HARD
40%
Subtopics
5

Strand: Long Tail

When you’ll see it

A count of ways — arrangements, selections, seatings, handshakes, or the lines and triangles determined by a set of points.

How this chapter is tested

40 q, 1.00/paper, 40% HARD, and it is the least mechanical chapter on the paper. No formula rescues a misread constraint: once the model is right the arithmetic is trivial, and when the model is wrong the arithmetic is worthless. That is why it is the single question most likely to eat five minutes of a 90-minute paper. Every PYQ is tagged to one of the five notes pages at /notes/mht-cet-maths/permutations-and-combinations.

The constraint questions — arrangements (9 q, 44% HARD), selections (7 q, 43%) and circular seatings (6 q, 83%) — run on a short recurring list: objects that stay together, objects never together, fixed positions, repeated letters, at least and at most. Each has one standard handling. Learning the handlings is far more productive than grinding assorted problems.

Because there is NO NEGATIVE MARKING, the discipline here is a time cap rather than a skip decision. Give the question ninety seconds; if the model has not resolved by then, mark the option whose order of magnitude matches your partial reasoning and move on. This is the chapter where that rule earns the most, because the downside of persisting is two or three other questions.

The cheaper corners are the identities page (8 q, none HARD) and the numbers-and-figures page (10 q, 40%): digit counts with a leading-zero exclusion, divisibility by the last digits or the digit sum, handshakes and diagonals as an nC2 equation, and triangles from points with the collinear picks subtracted. Three or four closed results cover them, so they are worth banking even though each is small.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Fundamental Principle, nPr and nCr — Definitions and Identities

    Multiply stages, add alternatives; nPr is nCr times r factorial; symmetry, Pascal's rule and the ratio of consecutive coefficients settle every equation-style stem; nPr and nCr are defined only for whole numbers n >= r >= 0.

  • Arrangements with Constraints — Together, Never Together, Fixed Positions and Repeated Letters

    Divide by k! per repeated letter; glue a together-group into a block and permute inside it; keep items apart by complement or by the gaps; fill a fixed position first and LIST the adjacent pairs that remain.

  • Selections with Conditions — At Least, At Most, Included and Excluded

    List the cases for at least and at most and add the products of nCr terms; subtract the forbidden selection when it is one simple case; multiply by the team size when a captain is chosen after the team.

  • Circular Arrangements

    n distinct people around a table give (n - 1)!; girls apart go into the b gaps between b boys (not b + 1); a glued block of k among n leaves (n - k)! times k!; alternating seats fix the frame.

  • Counting Numbers and Geometric Figures — Digits, Divisibility, Points and Polygons

    No leading zero; divisibility by 3 through the digit sum and by 25 through the last two digits; gcd conditions via inclusion-exclusion; nC2 handshakes and diagonals; nC3 triangles minus the collinear picks.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.

  • Order counted where it does not matter

    Using nPr for a selection inflates the answer by exactly r factorial. Both values are on the option list, which is why the inflated one is so easy to accept.

  • Block forgotten from the inside

    Grouping objects that must stay together and then failing to permute within the block undercounts by the factorial of the block size.

  • At least one, computed directly

    'At least one' is total minus none. Adding up the cases directly is slower and usually double-counts overlapping cases; the direct-sum answer is a supplied distractor.

  • Circular counted as linear

    n factorial instead of (n - 1) factorial for a round table. The linear value is offered, and it is exactly n times too large.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.

Permutations and Combinations notes

Drill every permutations and combinations question

40 questions from the bank, scoped to 5 bundled subtopics.

Drill the 40 questions

Drill one subtopic at a time

The 5 subtopics this playbook covers, in catalog order.

  • Fundamental Principle, nPr and nCr — Definitions and IdentitiesDrill
  • Arrangements with Constraints — Together, Never Together, Fixed Positions and Repeated LettersDrill
  • Selections with Conditions — At Least, At Most, Included and ExcludedDrill
  • Circular ArrangementsDrill
  • Counting Numbers and Geometric Figures — Digits, Divisibility, Points and PolygonsDrill

Related playbooks

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