Permutation and Combination Calculator

Permutations and combinations: A complete guide

Permutations and combinations are the backbone of counting and probability. They help you answer “How many ways?” in contexts like passwords, schedules, teams, lottery picks, and sampling. This guide explains both concepts clearly and shows when to use each, with and without repetition.

What are permutations?

Permutations count arrangements where order matters. Choosing positions for items or ranking contestants are classic permutation problems. For selections without repetition, the formula is \(nP_r=\frac{n!}{(n-r)!}\). With repetition allowed, each position can be any of the \(n\) items, so permutations become \(n^r\).

What are combinations?

Combinations count selections where order does not matter. Forming a committee or selecting a set of products are combination scenarios. Without repetition, the formula is \(nC_r=\frac{n!}{r!\,(n-r)!}\). With repetition, combinations become \(C(n+r-1,r)\), also written as \(\binom{n+r-1}{r}\).

Choosing the right model

• Order matters: Use permutations (rankings, seatings, codes with position significance).
• Order doesn’t matter: Use combinations (teams, sets, sample selections).
• Repetition allowed: Items can be reused (e.g., digits in a PIN, sampling with replacement).
• No repetition: Items cannot repeat (e.g., unique badges, sampling without replacement).

Common pitfalls and best practices

• Validate constraints: Without repetition, ensure \(r \le n\).
• Avoid decimal inputs: \(n\) and \(r\) must be non-negative integers.
• Match context: Decide if order and repetition apply before choosing a formula.
• Show working: Document formulas in reports for transparency.

Real-world applications

Use permutations for arranging shifts, route planning with distinct stops, and designing unique IDs. Use combinations for sample sizes, lottery odds, choosing features, and committee selection. In data science, these counts inform model complexity, feature selection, and probability calculations.

Worked examples