If you are wondering how to study Permutations and Combinations for JEE, start with one number. JEE Main asked 128 questions from Permutations and Combinations and Probability across 2024, 2025 and 2026: 63 from P&C and 65 from Probability. JEE Advanced asked 54 questions from the two chapters between 2015 and 2026, and 30 of those were numericals. This guide breaks all of them down, based on the papers conducted by the National Testing Agency (jeemain.nta.ac.in) for JEE Main and the JEE Advanced papers from 2015 to 2026.
We cover the two chapters together because they are one skill. Most Probability questions are counting questions with a division at the end, and P&C is where that counting is learned. We tagged every question by the counting idea or probability tool it needs. The tables below show exactly where the marks come from, and the rest of the guide shows how to win them.
These chapters have a reputation for being unpredictable. The papers say otherwise: a small number of question types repeat every year, often with only the numbers changed. Learn to recognise the type, and the method follows.
Together they are a big, steady block of marks. JEE Main had 46 questions from these two chapters in 2024, 42 in 2025 and 40 in 2026. That is about 7.7%, 8.8% and 8.4% of all Maths questions, or roughly two questions in every paper.
The balance is even. P&C gave 20, 21 and 22 questions in the three years, and Probability gave 26, 21 and 18. Neither chapter can be skipped in favour of the other.
They are numerical-heavy. 37 of the 128 Main questions were numericals, and 30 of the 54 Advanced questions were. There are no options to work backwards from, so your counting has to be exact.
JEE Advanced never skips them. Probability was asked in every Advanced paper from 2015 to 2026, and P&C in 11 of the 12 years. Counting also appears inside other chapters, from Matrices and Determinants (how many matrices have a given property) to relations and functions.
"Every counting question has one question hidden inside it: does order matter, and are the objects identical or different? Answer those two before you write a single formula, and most of the confusion in this chapter disappears." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
Real result · JEE 2026
P&C and Probability gave JEE Main 128 questions in three years, many of them numericals with no options to check against. Learn the counting types 1-on-1 with an IITian Maths HOD and stop losing marks to case splits.
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Any plan for how to study Permutations and Combinations for JEE should start from the papers. Here is how many questions JEE Main asked from each chapter in each of the last three years.
| Year | P&C | Probability | Total | MCQ + Numerical | Share of Maths Qs |
|---|---|---|---|---|---|
| 2024 | 20 | 26 | 46 | 31 + 15 | 7.7% |
| 2025 | 21 | 21 | 42 | 32 + 10 | 8.8% |
| 2026 | 22 | 18 | 40 | 28 + 12 | 8.4% |
| Total | 63 | 65 | 128 | 91 + 37 | — |
Here are the 63 P&C questions sorted by type.
| P&C type | 2024 | 2025 | 2026 | Total |
|---|---|---|---|---|
| Arrangements: words, numbers with conditions, queues, dictionary rank | 5 | 10 | 8 | 23 |
| Selections: committees, restricted choices, counting triangles and polygons | 5 | 6 | 5 | 16 |
| Distribution: objects into groups or boxes, integer solutions, digit sums | 7 | 2 | 5 | 14 |
| Counting functions and inclusion–exclusion | 0 | 2 | 1 | 3 |
| Other (ⁿCr algebra, the power of a prime in n!) | 2 | 1 | 3 | 6 |
| Not classified* | 1 | 0 | 0 | 1 |
| Total | 20 | 21 | 22 | 63 |
And here are the 65 Probability questions sorted by the tool that solves them.
| Probability type | 2024 | 2025 | 2026 | Total |
|---|---|---|---|---|
| Classical probability by counting (dice, coins, numbers, words, matrices) | 8 | 7 | 8 | 23 |
| Random variables: distributions, mean and variance, binomial and repeated trials | 8 | 5 | 4 | 17 |
| Total probability and Bayes' theorem | 5 | 5 | 6 | 16 |
| Conditional probability and independence | 4 | 4 | 0 | 8 |
| Other chapter (sets) | 1 | 0 | 0 | 1 |
| Total | 26 | 21 | 18 | 65 |
*The archived text of one 2024 P&C question did not render fully. It is counted in the totals only. One 2026 question listed with Probability in the archive but filed under Statistics is not counted.
The message of these tables is simple. In P&C, arrangements and selections gave 39 of 63 questions. In Probability, counting-based questions gave 23 of 65, and Bayes' theorem with total probability gave another 16, one in every four. Random variables, often neglected, gave 17.
Pattern 1: Words and numbers with a condition. 4-letter words from INCONSEQUENTIAL with two vowels and two consonants (2026), 4-letter words from DISTRIBUTION (2024), 7-digit numbers using 1, 2, 3, 5 and 7 with every digit at least once (2026), 7-digit numbers from 1, 2 and 3 with digit sum 11 (2025). The trick is always to split into cases by the repeated letters or digits.
Pattern 2: Dictionary rank. UDAYPUR (2026), KANPUR and the 440th word (2025), NAGPUR, BHBJO and GTWENTY (2024). Three years of papers, six dictionary questions. The method never changes: count the words that start with each smaller letter, then move one position right.
Pattern 3: Identical objects and integer solutions. 16 oranges to 4 children with at least one each (2026), 21 apples to 3 children with at least two each (2024), a + b + 2c = 22 in non-negative integers (2026), integers between 100 and 1000 with digit sum 14 (2024). All are stars and bars, sometimes after a substitution.
Pattern 4: Bayes with coins, bags and plants. 19 fair coins and one two-headed coin (2025), N fair coins and one two-headed coin (2026), three bags, urns or plants with a defect rate (2024 five times). Draw the tree, write P(cause) × P(effect | cause) on each branch, and divide.
Pattern 5: Variance of a random variable. The number of defective items or rotten apples in a sample appeared five times from 2024 to 2025, and a binomial "at least 7 of 8 defective" question in 2026. Write the probability table first; the mean and variance then come straight from it.
This is the 12-year view that most guides skip. JEE Advanced asked Probability in every year from 2015 to 2026, and P&C in every year except 2023.
| Year | P&C | Probability | What was asked |
|---|---|---|---|
| 2026 | 2 | 2 | Equivalence relations on a 10-element set · 10 red and 14 blue identical pens to 4 persons, 6 each · independence of events from two pooled boxes · the mean of a difference of counts |
| 2025 | 2 | 2 | Reflexive and symmetric relations on a 4-element set · 7-digit numbers with a digit used exactly twice · three students attempting a problem · defect rates of three units (Bayes) |
| 2024 | 3 | 3 | Two numericals on counting relations and functions · 9 students into teams of 2, 3 and 4 · a true/false quiz with guessing (Bayes) · a bag of N balls · a random variable with given mean and variance |
| 2023 | 0 | 5 | Tossing until two consecutive tosses match · lattice triangles with integer area · multiples of 20 among multiples of 5 · two questions on "friend" points |
| 2022 | 2 | 3 | 4-digit integers in [2022, 4482] · 10 balls from 4 boxes · a chain of boxes (Bayes) · dice rounds with scoring (match) · 900 persons with overlapping symptoms |
| 2021 | 1 | 5 | Sizes of sets of tuples · random picks between sets · bounds on event probabilities · multiples of 3 or 7 up to 2000 · two numericals on the maximum and minimum of three numbers |
| 2020 | 2 | 3 | 4 visits in 15 days with no two consecutive · 6 persons in 4 rooms · equal roots from coin tosses · the minimum number of missiles · rolling until a prime or square sum |
| 2019 | 2 | 2 | Ordered pairs of independent events · hats around a circle with adjacent colours different · three bags (Bayes) · 3 × 3 matrices with det = 0 given the entry sum |
| 2018 | 2 | 2 | Committees from 6 boys and 5 girls (match) · 5-digit numbers divisible by 4 · two paragraph questions on reseating five students |
| 2017 | 1 | 2 | Words of length 10 with one repeated letter · x + y + z = 10 with z even · statements on conditional probabilities |
| 2016 | 1 | 3 | A team with a captain and at most one boy · two paragraph questions on two teams' points · two plants (Bayes) |
| 2015 | 1 | 3 | Girls together in a queue · two multi-correct questions on two boxes (Bayes) · the minimum number of tosses for two heads |
| Total | 19 | 35 | 30 numerical · 11 single correct · 7 multi-correct · 4 paragraph · 2 match |
One 2019 question filed under P&C (ordered pairs of independent events) is really a Probability question. It is counted under P&C, where the archive files it.
Advanced has two clear differences from Main. In P&C, counting relations and functions gave 4 of the 19 questions, all from 2024 to 2026, so it is a new and growing type. In Probability, the questions are longer and more often built on a story: a tournament, a quiz with guessing, a chain of boxes. Probability gave 10 questions on random variables and repeated trials, 9 each on classical and conditional probability, and 7 on Bayes' theorem.
The smartest way to decide how to study Permutations and Combinations for JEE, and Probability alongside it, is to match your time to each type. Use the time column in your mock tests: if you are consistently slower than the target, that type needs more practice, not more reading.
| Type | Main Qs (2024–26) | Advanced Qs (2015–26) | Difficulty | Target time |
|---|---|---|---|---|
| P&C: arrangements and dictionary rank | 23 | 5 | Medium | 2–3 min |
| Probability: classical (counting) | 23 | 9 | Medium | 2–3 min |
| Probability: random variables and binomial | 17 | 10 | Medium | 2–3 min |
| P&C: selections and geometry counting | 16 | 5 | Easy–Medium | 90 s–2 min |
| Probability: total probability and Bayes | 16 | 7 | Easy–Medium | 2 min |
| P&C: distribution and integer solutions | 14 | 3 | Medium–Hard | 2–3 min |
| Probability: conditional and independence | 8 | 9 | Medium | 2 min |
| P&C: counting functions and relations, inclusion–exclusion | 3 | 4 | Hard | 3 min |
| P&C: circular arrangements | 0 | 1 | Medium | 2 min |
Difficulty and target time are our faculty's guidance for exam practice, not exam data.
Arrangements covers permutations of distinct and repeated objects, numbers formed from digits with conditions (divisibility, digit sums, range), people in a row with "together" or "never together" conditions, and dictionary rank.
Selections covers combinations with restrictions (at least, at most, must include), committees from groups, and geometry counting: lines, triangles, polygons and diagonals from given points.
Distribution covers identical objects into distinct boxes (stars and bars), distinct objects into distinct or identical boxes, and integer solutions of equations with bounds.
Classical probability is favourable outcomes divided by total outcomes. Every counting skill from P&C is used here.
Total probability and Bayes covers problems where an outcome can come from several causes (bags, coins, machines) and you either want the overall probability or the probability of a cause given the outcome.
Random variables covers probability distributions, mean and variance, the binomial distribution and "repeat until" problems.
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Bayes' theorem, random variables and counting-based probability gave 56 of 65 JEE Main Probability questions. Work through them with your own faculty, at your pace, with every doubt answered the same day.
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Knowing how to study Permutations and Combinations for JEE also means knowing what not to do. These are the mistakes we see most often when students solve these chapters' PYQs.
Keep this table open while you practise. Most of the two chapters run on these results, which is why learning how to study Permutations and Combinations for JEE is mostly about choosing the right formula for the situation.
| Sub-topic | Formula / result |
|---|---|
| Basics | ⁿPr = n!/(n − r)! · ⁿCr = n!/(r!(n − r)!) · ⁿCr = ⁿCn−r · ⁿCr + ⁿCr−1 = ⁿ⁺¹Cr |
| Repeated objects | n objects with p alike, q alike, …: n!/(p! q! …) · r-letter words from a multiset: split into cases by repeats |
| Circular | n distinct objects: (n − 1)! · necklace or garland (flip allowed): (n − 1)!/2 · colouring a cycle of n with k colours, adjacent different: (k − 1)n + (−1)n(k − 1) |
| Together / apart | Keep k items together: treat as one block, multiply by k! · no two of m items together: arrange the others, choose m of the gaps |
| Stars and bars | x1 + … + xr = n, xi ≥ 0: ⁿ⁺ʳ⁻¹Cr−1 · xi ≥ 1: ⁿ⁻¹Cr−1 · lower bounds: subtract them first |
| Distinct into distinct | n distinct objects into r distinct boxes: rn · none empty: rn − C(r, 1)(r − 1)n + C(r, 2)(r − 2)n − … |
| Grouping | Divide n distinct objects into groups of sizes a, b, c (unequal): n!/(a! b! c!) · equal groups: divide also by (number of equal groups)! |
| Geometry counting | n points, m collinear: lines ⁿC2 − ᵐC2 + 1, triangles ⁿC3 − ᵐC3 · diagonals of an n-gon: n(n − 3)/2 · triangles from n-gon vertices with no side common: n(n − 4)(n − 5)/6 |
| Derangements | Dn = n!(1 − 1/1! + 1/2! − … + (−1)n/n!) · D3 = 2, D4 = 9, D5 = 44 |
| Functions and relations | A → B with |A| = m, |B| = n: all nᵐ, one-one ⁿPm · relations on a set of n elements: 2n² · reflexive: 2n² − n · symmetric: 2n(n+1)/2 |
| Prime in n! | Power of prime p in n! = ⌊n/p⌋ + ⌊n/p2⌋ + ⌊n/p3⌋ + … |
| Probability rules | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) · P(A | B) = P(A ∩ B)/P(B) · independent: P(A ∩ B) = P(A)P(B) |
| Total probability and Bayes | P(E) = Σ P(Ai)P(E | Ai) · P(Ak | E) = P(Ak)P(E | Ak)/Σ P(Ai)P(E | Ai) |
| Random variables | Mean μ = Σ x p(x) · variance σ2 = Σ x2p(x) − μ2 · Σ p(x) = 1 |
| Binomial | P(X = r) = ⁿCr prqn−r · mean np · variance npq · "at least one" = 1 − qn |
| Repeat until | First success on trial k: qk−1p · sums of such terms are geometric series: a/(1 − r) |
This is the core of how to study Permutations and Combinations for JEE, and Probability after it: P&C types first, because Probability uses them, then the probability tools. Do not move on until you can solve the PYQ types listed under each heading without notes.
The largest P&C group. Three habits make it reliable:
Worked example from JEE Main 2026: 4 boys and 3 girls stand in a queue so that the girls are not all together. All arrangements: 7! = 5040. Girls all together: treat them as one block, so 5! × 3! = 720. The answer is 5040 − 720 = 4320.
Worked example from JEE Main 2026: 7-digit numbers using the digits 1, 2, 3, 5 and 7, each at least once. This is the number of onto functions from 7 positions to 5 digits: 57 − 5 × 47 + 10 × 37 − 10 × 27 + 5 × 17 = 78125 − 81920 + 21870 − 1280 + 5 = 16800.
Practice target: 30 arrangement PYQs, including every dictionary-rank question from 2024 to 2026.
Selections are the easiest marks once you split cases cleanly.
Worked example from JEE Main 2026 (numerical): in how many ways can A win a series that ends when one player has won 5 games, with no draws? A wins the last game and 4 of the earlier ones, while B wins k games (k = 0 to 4). The count is the sum of ⁴⁺ᵏCk = 1 + 5 + 15 + 35 + 70 = 126.
Practice target: 20 selection PYQs. Write the case table before calculating each one.
This group decides between strong and average scores, because each problem needs the right one of four models: identical or distinct objects, into identical or distinct boxes.
Worked example from JEE Main 2026: count non-negative (a, b, c) with a + b + 2c = 22. For each c from 0 to 11, a + b = 22 − 2c has 23 − 2c solutions. The total is 23 + 21 + … + 1 = 144.
Practice target: 15 distribution PYQs, labelled by which of the four models each uses.
"In distribution problems, the first line of your solution should be four words: identical or distinct objects, identical or distinct boxes. Once that is written, the method is fixed. Students who skip that line pick a formula by habit and get it wrong." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
Count the total outcomes, count the favourable ones, divide. All the P&C skills above apply.
Worked example from JEE Main 2026: two distinct numbers a and b are chosen from 1 to 50. What is the probability that ab is divisible by 3? Use the complement. There are 34 numbers not divisible by 3, so P(neither is a multiple of 3) = ³⁴C2/⁵⁰C2 = 561/1225, and the answer is 664/1225.
Worked example from JEE Main 2024 (numerical): a tetrahedral die (faces 1 to 4) is rolled three times to give a, b and c. P(ax2 + bx + c = 0 has real roots) needs b2 ≥ 4ac. Counting: b = 2 gives 1 case, b = 3 gives 3 and b = 4 gives 8, so 12 of 64 outcomes. The probability is 3/16 and m + n = 19.
Practice target: 25 classical PYQs, always writing the size of the sample space first.
Draw a tree with the causes on the first branches and the outcome on the second.
Worked example from JEE Main 2025: a bag has 19 fair coins and one two-headed coin. A coin is drawn and tossed, and shows a head. P(fair) = (19/20 × 1/2)/(19/20 × 1/2 + 1/20 × 1) = (19/40)/(21/40) = 19/21.
Worked example from JEE Main 2026: the probability that A is captain is 0.6 and B is 0.4, and the team wins with probability 0.8 or 0.7 respectively. By total probability, P(win) = 0.6 × 0.8 + 0.4 × 0.7 = 0.76.
Practice target: 20 Bayes PYQs. Draw the tree every time, even for two branches.
Random variables are the quiet scorer of these chapters, and conditional probability links them to everything else.
Practice target: 20 PYQs from these two groups, including every variance question from 2024 and 2025.
That is how to study Permutations and Combinations for JEE, and Probability with it, type by type. Once you have finished, test yourself under exam conditions with our JEE test series, and bring any question that still does not make sense to the JEE Doubt Lab for a 1-on-1 explanation.
Ready for exam-level practice?
JEE Advanced asks these chapters every year, mostly as numericals: counting relations, story-based probability and minimum-trial problems. Practise them 1-on-1 with an IITian Maths HOD.
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If you are deciding how to study Permutations and Combinations for JEE Advanced as well as Main, this section is for you. Advanced goes further in four places.
From 2024 to 2026, Advanced asked four numericals that count relations or functions: equivalence relations on a 10-element set with a condition (2026), reflexive and symmetric relations on {a, b, c, d} (2025), and two linked questions on relations and functions (2024). Learn the counting formulas for reflexive, symmetric and equivalence relations, and the link between equivalence relations and partitions of a set.
Advanced 2020 asked for the ways to visit a factory on exactly 4 of the first 15 days of a month with no two visits on consecutive days. Choosing 4 non-adjacent days from 15 is ¹²C4 = 495, by the gap method. Advanced 2019 asked for hat colourings of five people round a table with adjacent hats different, using 3 colours: (3 − 1)5 + (−1)5(3 − 1) = 30.
Advanced 2020 asked how many missiles, each hitting with probability 0.75, are needed so that the probability of at least 3 hits is at least 0.95. With 5 missiles the probability is about 0.90; with 6 it is about 0.96, so the answer is 6. Advanced 2015 asked a similar question for coin tosses. Compute the complement for successive n until it crosses the threshold.
Advanced likes probability set inside a process: two teams playing games with points (2016), dice rounds with scoring rules (2022), tossing until two consecutive tosses match (2023), and random picks moving between three sets (2021). Advanced 2017 asked for the probability that z is even when x + y + z = 10 is solved in non-negative integers: 36 of the 66 solutions, so 6/11. Break each story into stages and write the probability of each stage before combining.
"Advanced probability questions are long because they tell a story. Turn the story into a tree or a table on paper, then the probability is arithmetic. Do not try to hold the whole story in your head." — AKS Yadav Sir, Maths HOD (IIT BHU, AIR 1 FTRE 2009, Ex-Faculty Allen Delhi)
The last step in how to study Permutations and Combinations for JEE, and Probability, is an honest self-check. You should be able to answer "yes" to every line below without notes.
Permutations and Combinations and Probability gave JEE Main 128 questions from 2024 to 2026 (63 and 65) and JEE Advanced 54 from 2015 to 2026 (19 and 35), with Probability asked in every Advanced paper. In Main, arrangements and selections gave 39 of the P&C questions, and counting-based probability, Bayes' theorem and random variables gave 56 of the Probability questions. In Advanced, counting relations is the new type and story-based probability is the hard one. So if you are working out how to study Permutations and Combinations for JEE, the order is clear: arrangements and selections first, then distribution, then classical probability and Bayes, then random variables, and finish with relations and story problems for Advanced.
How we built this data. We went through every JEE Main shift paper from 2024, 2025 and 2026 and every JEE Advanced paper from 2015 to 2026, picked out each question from Permutations and Combinations and from Probability, and tagged it by the method that solves it. Where the archived text of a question was too incomplete to classify with confidence, we counted it in the totals and said so instead of guessing.
If you want a faculty member to check your preparation, every pack starts with a free 30-minute mentorship call. Permutations & Combinations and Probability each have their own packs on our JEE Maths coaching page, and once your exam is done, our JEE counselling team can help you turn your rank into the right college and branch.
Counting all shifts, JEE Main asked 46 questions from the two chapters in 2024, 42 in 2025 and 40 in 2026. P&C gave 20, 21 and 22 of those, and Probability gave 26, 21 and 18. That is about two questions in a typical paper.
It feels difficult because a small slip in the case split changes the answer, and many questions are numericals with no options to check against. With practice on the standard types it becomes steady. If you follow this guide on how to study Permutations and Combinations for JEE, the repeating question types become familiar quickly.
Yes. They gave 128 JEE Main questions from 2024 to 2026, and Probability was asked in every JEE Advanced exam from 2015 to 2026, with P&C in 11 of those 12 years.
No. Skipping it gives up about one question per JEE Main paper directly, and it also makes most Probability questions impossible, because they are counting questions with a division at the end.
In P&C, master arrangements and selections: they gave 39 of the 63 JEE Main questions from 2024 to 2026, and adding distribution covers 53. In Probability, classical probability, Bayes' theorem and random variables gave 56 of 65.
Counting relations and functions (4 Advanced P&C questions from 2024 to 2026), the gap method and circular colouring, minimum-number-of-trials problems, and long story-based probability questions. 30 of the 54 Advanced questions from 2015 to 2026 were numericals.
