Playbook
Permutations and Combinations
43 q - 1.00/paper - 42% HARD. The least mechanical chapter on the paper: no formula rescues a misread constraint. Selection and Arrangement with Constraints is 33 of its 43 q. One question a paper, and the one most likely to eat five minutes.
- Questions in the bank
- 43
- q/paper (2024–25 shifts)
- 1.00
- Tagged HARD
- 42%
- Subtopics
- 2
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
43 q, 1.00/paper, 42% 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.
Selection and Arrangement with Constraints carries 33 of the 43 questions at 42% HARD. The constraints themselves are a short recurring list — certain objects must stay together, certain objects must never be together, some positions are fixed, repetition is or is not allowed — and each has one standard handling. Learning the four 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.
Counting and Geometric Applications (10 q, 40% HARD) is the narrower and more mechanical corner: lines and triangles from n points with collinear subsets subtracted, diagonals of a polygon, and similar. Three or four closed results cover it, so it is worth banking even though the subtopic is small.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Fundamental counting principle
Multiply when choices happen in sequence and every stage is required; add when the cases are alternatives that cannot both happen. Deciding add-versus-multiply is the first and most consequential step.
Permutation versus combination
Order matters means nPr; order does not means nCr; and nPr is nCr times r factorial. Almost every over-count in this chapter is a permutation used where a combination belonged.
Repetition and identical objects
Arrangements of n objects of which some are identical divide n factorial by the factorials of the repeat counts. Arrangements with unlimited repetition over r places from n symbols are n^r.
Constraint handling
Objects that must stay together: glue them into one block, arrange the blocks, then permute inside the block. Objects that must never be together: arrange the rest first, then place them in the gaps. Fixed positions: fill those first and count what is left.
Circular arrangements
n distinct objects around a circle give (n - 1) factorial, because rotations are the same arrangement. Halve it again when reflections also count as the same, such as an unmarked necklace.
Geometric counting
From n points with no three collinear: nC2 lines and nC3 triangles. When m of them ARE collinear, subtract mC2 lines and mC3 triangles and add one line back.
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.
Drill every permutations and combinations question
43 questions from the bank, scoped to 2 bundled subtopics.
Drill one subtopic at a time
The 2 subtopics this playbook covers, in catalog order.
Related playbooks
Often paired with this one — the technique, the trap or the taxonomy overlaps. Drill these next.