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Learn Combinatorics

Count anything without listing it, from poker hands to passwords.

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Combinatorics is the mathematics of counting, arranging and choosing. How many ways can 8 people sit in a row? How many five-card hands contain a pair? How many routes cross a grid? The answers come from a small toolkit: the multiplication principle, permutations, combinations, complementary counting, stars and bars, inclusion-exclusion, the pigeonhole principle and recurrences. It underlies probability, computer science and competition math, and it rewards careful thinking more than heavy computation. Learnr builds a combinatorics course around your level and teaches it through short interactive lessons where you predict a count before checking it.

How to learn Combinatorics, step by step

Ground everything in the multiplication principle. Count license plates, outfits and passwords by multiplying the choices at each stage. Then learn when to add instead, when the cases cannot happen together.
Separate order-matters from order-does-not. Arranging 3 of 5 books on a shelf gives 5 times 4 times 3, or 60. Choosing 3 of the same 5 to pack gives 60 divided by 3 factorial, which is 10. Ask whether order matters every single time.
Check small answers by listing. For each formula, test a tiny case by writing out every outcome. Listing the 10 ways to choose 2 letters from ABCDE catches double counting faster than any rule.
Learn stars and bars and complements. Distributing 10 identical candies to 3 children gives C(12, 2), or 66, ways. For "at least one" questions, count the total and subtract the cases with none.
Add inclusion-exclusion and pigeonhole. Count numbers up to 100 divisible by 2 or 3 as 50 plus 33 minus 16, giving 67. Then use pigeonhole arguments to prove that some outcome must repeat.
Solve a few problems every day. Counting skill grows from variety. A short Learnr session daily, with a streak to keep you honest, exposes you to enough problem types that the right technique starts to jump out.

What you'll learn

Apply the addition and multiplication principles correctly
Choose between permutations and combinations for any problem
Count with stars and bars and complementary counting
Use inclusion-exclusion and the pigeonhole principle
Set up recurrences and use the binomial theorem

What's inside your Combinatorics course

A sample path Learnr builds for Combinatorics. Your real course adapts to your goal and level.

1. Addition and multiplication principles
2. Permutations, combinations and factorials
3. Pascal's triangle and the binomial theorem
4. Stars and bars and distributions
5. Inclusion-exclusion and the pigeonhole principle
6. Recurrences, Catalan numbers and generating functions

How long does it take to learn Combinatorics?

Core counting techniques take about 4 to 8 weeks at 3 to 5 hours a week. Reaching competition-level combinatorics, with generating functions and clever bijections, usually takes 6 months to a year of regular problem solving.

What you need before you start

Solid arithmetic, comfort with exponents and basic algebra. No calculus is needed; probability and discrete math courses often teach combinatorics alongside.

Common mistakes to avoid

Using permutations when order does not matter, which overcounts by a factor equal to the number of ways to arrange the chosen items.
Double counting by splitting into cases that overlap. Make sure every outcome lands in exactly one case before adding.
Trusting a formula without testing a small case by hand, which is the quickest way to catch an off-by-one error.

How Learnr builds your course

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Combinatorics questions, answered

What is the difference between a permutation and a combination?

A permutation counts arrangements where order matters, like race finishes. A combination counts selections where order does not, like a committee. Combinations equal permutations divided by the number of orderings of the chosen items.

Is combinatorics needed for probability?

Yes, for any problem with equally likely outcomes. The probability is the count of favorable outcomes divided by the count of all outcomes, and combinatorics supplies both counts.