LearnViz CompSci
Home / Learn / Theory & Discrete Math / Permutations and Combinations

Beginner Topic 8 of 51 Theory & Discrete Math

Permutations and Combinations

Counting ordered and unordered selections.

How it works

A permutation is an ordered arrangement of items, and the number of ways to arrange r of n distinct items is n!/(n-r)!. A combination is an unordered selection, counted by the binomial coefficient n choose r, equal to n!/(r!(n-r)!). Variants include permutations with repeated elements, divided by factorials of repeat counts, and combinations with repetition, counted by stars and bars. Pascal's triangle and the binomial theorem connect them.

Step-by-step walkthrough

  1. Permutations count ordered selections, n! over (n-k)!
  2. Combinations ignore order, n! over k!(n-k)!
  3. Choosing 2 of 4 people in order gives 12.
  4. Without order it gives 6, each pair counted twice.
  5. Use permutations for rankings and combinations for teams.

Where it is used

Counting passwords and lottery outcomes, scheduling, generating test cases, probability calculations and algorithm analysis, such as the number of subsets 2 to the n, use these formulas.

Common pitfall

Choosing between order mattering and not mattering is the main difficulty. Factorials grow explosively, so brute-force enumeration quickly becomes infeasible. Overflow occurs when computing factorials directly; use iterative multiplication and division or Pascal's recurrence. Check the problem for identical items and restrictions.

Where it fits in the course

Permutations and Combinations is topic 8 of 51 in Theory & Discrete Math, rated Beginner. It follows Counting Principles. The next topic is Pigeonhole Principle.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Theory & Discrete Math quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Permutations and Combinations?
Counting ordered and unordered selections. It belongs to Theory & Discrete Math and is a beginner-level topic.
Where do people use Permutations and Combinations?
Counting passwords and lottery outcomes, scheduling, generating test cases, probability calculations and algorithm analysis, such as the number of subsets 2 to the n, use these formulas.
What is the most common mistake with Permutations and Combinations?
Choosing between order mattering and not mattering is the main difficulty.
What should I learn before this topic?
Start with Counting Principles, the topic just before it in Theory & Discrete Math.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Theory & Discrete Math quiz.
Is Permutations and Combinations hard to learn?
It is a Beginner topic, so it works as an entry point. If the terms are new, read the explanation twice and try the visualizer before the quiz.
How long does it take to learn Permutations and Combinations?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Permutations and Combinations?
Counting Principles, Pigeonhole Principle, Proof Techniques are closely connected. The full list is in the related topics section of this page.
What should I study after Permutations and Combinations?
Continue with Pigeonhole Principle, the next topic in Theory & Discrete Math.

Related topics