PYQ Vault

Playbook

Permutations and Combinations

More of it is numeric-answer than any other chapter still on the paper, so there are no options to catch a slip. Own it outright.

Questions in the bank
160
q/paper in 2025–26
1.22
Numeric answer
58%
Notes pages
8

Tier: Core

When you’ll see it

A count of arrangements, selections, numbers built from digits, distributions or figures formed by points, usually with a restriction.

How this chapter is tested

Much of the chapter is set as numeric-answer questions, where there are no options to catch a slip. Forming numbers from digits, selections and counting divisors carry most of it; each is a case split followed by a product of simple counts.

The first decision is the whole question: does order matter, are the objects identical, are the boxes distinct. Get it wrong and every later step can be right while the answer is wrong. For 'at least' or 'not' conditions, count the complement.

The chapter feeds the binomial theorem, probability, and relations and functions: counting functions, matrices and subsets uses the same position-by-position product. The rank of a word in dictionary order is one fixed procedure and is quick to score.

The sub-skills

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

  • Arrangements with restrictions

    Blocks for 'together', gaps for 'apart', (n − 1)! around a table, n!/(p! q!) for repeated items.

  • Dictionary rank

    Go letter by letter and count the words that start with a smaller letter; recompute the divisor when a repeated letter is used up.

  • Forming numbers from digits

    Fill the restricted places first; 0 cannot lead; split by the last digit for divisibility conditions.

  • Selections and committees

    nCr for choices; split 'at least' conditions into exact cases, or use the complement.

  • Distributions and counting objects

    n identical objects into r distinct boxes: (n + r − 1)C(r − 1); distinct objects by onto maps; functions, matrices and subsets filled position by position.

  • Points, lines and polygons

    Choose vertices and subtract collinear triples; an n-gon has nC2 − n diagonals.

  • Divisors and factorials

    The power of a prime p in n! is [n/p] + [n/p²] + …; divisors from prime powers; multiples in a range by inclusion–exclusion.

Traps to expect

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

  • 'Not all together' is not 'no two together'

    'The vowels never all together' is the total minus the all-together case and allows two vowels side by side. 'No two vowels together' uses the gap method.

  • Zero in the leading place

    When 0 is available, the first digit has one fewer choice. If the last digit must be even, split into 'ends in 0' and 'does not'.

  • Choosing 'at least one' first

    Picking one woman and then any others counts the same committee several times. Split into exact cases instead.

  • Product for the overlap

    Numbers divisible by both 4 and 6 are the multiples of 12, the lcm, not of 24.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Permutations and Combinations notes

Drill every Permutations and Combinations question

160 questions from the bank, across 8 subtopics.

Drill the 160 questions

Drill one subtopic at a time

The 8 subtopics, in teaching order.

Related playbooks

Often paired with this one — the technique or the trap overlaps. Drill these next.