Combinatorics Interview Questions
Combinatorics is the art of counting structured sets without enumerating them one by one, and it is the bread-and-butter of brainteaser rounds at every quant shop. This playlist builds the core toolkit from the ground up: the multiplication and addition principles, permutations and combinations, mul
How to think about combinatorics questions
Counting problems reward laziness of the right kind: instead of enumerating, you find a structure that does the counting for you. A bijection, a symmetry, or a clever complement turns a hopeless tally into a one-liner.
COUNT IT TWICE
Set up a quantity you can count two different ways and force the answers to agree — the heart of every binomial identity and stars-and-bars argument. When the front door is blocked, complementary counting (total minus the bad cases) is often wide open.
INCLUDE, THEN EXCLUDE
Overlapping conditions are the usual source of pain. Inclusion–exclusion says: add everything, subtract the double-counted pairs, add back the triples — an alternating correction that makes “at least one” problems tractable without case soup.
The recurring instinct: don't list the objects — find a map, a symmetry, or a complement that counts them for you.
Combinatorics questions (86)
- Ordered Triples Summing to 20
- Positive Integer Solutions via Stars and Bars
- Portfolio Allocation via Stars and Bars
- Counting Topping Combinations
- Arranging Colored Balls in a Line
- Grid Path Counting
- Grid Paths With a Checkpoint
- VIP Room Handshakes
- Guaranteed Birth Month Match: Pigeonhole Principle
- Counting Proper Parenthesizations (Catalan Numbers)
- Guest List with Exclusion Constraint
- Sock Drawer Guarantee
- Compositions of an Integer
- Binary Strings Starting With 1 or Ending With 00
- Counting Non-Decreasing Digit Strings
- Counting Squares in an N x N Grid
- Dividing Students into Labeled Groups
- Contiguous Block Permutations in a Race
- Three-State Subset Count with Bold Elements
- Subsets with Exactly Two Elements from a Restricted Range
- Jenga Stack
- Counting Sub-Cubes Inside a Cube
- Unpainted Cubes Inside a Painted 10x10x10 Cube
- Three-Digit Codes with Bounded Digit Sum
- Counting Distinct Age Combinations
- Counting Positive Divisors via Prime Factorization
- Binary Search Guessing Game
- Distinct Marbles Across Bags
- 100 Switches and Perfect Squares
- Arranging Chocolate and Mint Sticks
- Binomial Coefficient Symmetry Match
- Finding the Heavy Bean With a Balance Scale
- Single-Elimination Tournament Bracket Count
- Coefficient of a Specific Term in a Binomial Expansion
- Count Even-Product Triplets
- Trailing Zeros of 100 Factorial
- Alternating Sum of Binomial Coefficients
- Smallest Integer With Consecutive Remainders
- Count Integers with Distinct Digits
- Permutations With No Adjacent Evens
- MISSISSIPPI Anagrams Starting and Ending with the Same Letter
- Arrangements of LLLEEEEWIS with Two L's Before the First E
- Circular Seating Arrangements From a Group
- Distributing Balls Into Urns With Fixed Capacity
- Counting Integers With a Given Digit Sum
- Distinct Balls in Identical Bins
- Identifying a Poisoned Cake With Binary Encoding
- Balance Scale and Weights
- Counting Paths to Deuce in a Tennis Game
- Minimum Tests to Find the Broken Revision
- Minimum Pairs to Guarantee a Modular Match
- Distinct Orders to Break Stacked Targets
- Shuffle Permutation Order for 14 Cards
- Finding the Top Three Horses in Minimum Races
- Distinct Labelings of a Die Under Rotation
- Counting Odd-Only Subsets with One Element from a Prefix
- Alternating Even-Odd Permutations of [10]
- Matchy Palindromes
- Count Integers with All Distinct Digits
- Sum of All Digits from 1 to One Million
- Grid Paths With At Least One Power-Up
- Round-Robin Tournament Point Distribution
- Josephus Problem: Last Person Standing in a Circle
- Counting Almost Square Numbers
- Battleship on a 4x4 Board: Arrangements and Optimal Targeting
- Chess Club Photo Arrangement
- Twelve-Digit Numbers with Prime Digits and Product One Million
- Sum of Weights From Pairwise Sums
- Counting Valid Arithmetic Expressions with Digits 1-9
- Sharing a Secret: Locks and Keys
- Pirate Ship Cannons
- Sicherman Dice: Non-Standard Dice With the Same Sum Distribution
- Counting Distinct Bracelets with 3 Red and 3 Green Beads
- Counting Squares on a Grid
- Uniform Sum Distribution With Relabeled Dice
- Total Score of a Random Partitioning Game
- Counting Digit Occurrences in a Range
- Limit of Integer Partition Count
- Monotone Signaling via Breaking Glass
- Max Sum Pairwise Coprime Subset
- Recover Individual Watermelon Weights from Pairwise Sums
- Medal Assignments with Ties
- Largest Fancy Number At Most N
- Arithmetic Progression-Free Sequence Completion
- Enchanted Forest Tree Toggling
- Minimum Clues for a Unique Sudoku
Combinatorics interview questions FAQ
What kind of combinatorics questions show up in quant interviews?
This page collects 86 combinatorics problems that recur in quant trading and research interviews, each with a full worked solution and the intuition behind it. They range from quick warmups to the harder variants firms use to separate candidates.
How hard are combinatorics interview questions?
The set spans 36 easy, 38 medium and 12 hard problems. Most sit at medium difficulty — a few minutes of clean reasoning — with a harder tail that rewards knowing the canonical approach rather than grinding.
How should I practice combinatorics for quant interviews?
Work through them by difficulty, starting just below your level, and write the solution out before checking. 15 are free to open with the full worked solution, so you can judge the quality first. Focus on the recurring patterns rather than memorizing answers — the same handful of ideas generate most variants.
Are these real quant interview questions?
They are a curated set drawn from our problem bank — the kind of combinatorics question that actually appears in quant interviews, rewritten for clarity with solutions we author ourselves. We don't claim any single wording is verbatim, and every problem carries a full solution.