Combinations & Permutations Calculator

This calculator determines the number of ways to select and arrange elements from a set, distinguishing between permutations (order matters) and combinations (order does not matter).

Formulas

The fundamental formulas are:

• Permutations P(n,r): P(n,r) = n! / (n−r)!
• Combinations C(n,r): C(n,r) = n! / (r! × (n−r)!)

Where n is the total number of elements and r is the number of elements selected. The factorial n! = n × (n−1) × ... × 2 × 1.

Example 1: permutations

Problem: In how many ways can you arrange 3 books from a shelf of 8?

1. Calculation:
   • P(8,3) = 8! / (8−3)! = 8! / 5! = 8 × 7 × 6 = 336.

Answer: 336 different arrangements.

Example 2: combinations

Problem: In how many ways can you choose 3 people from a group of 8?

1. Calculation:
   • C(8,3) = 8! / (3! × 5!) = (8 × 7 × 6) / (3 × 2 × 1) = 56.

Answer: 56 different groups.

Common uses

• Computing probabilities in games of chance and lotteries.
• Designing experiments and statistical sampling.
• Analyzing possibilities in optimization problems.
• Solving probability problems in education.
• Computing possible passwords in computer security.
• Planning schedules and resource assignments.

Common mistakes

• Using combinations when order matters (should be permutation).
• Using permutations when order does not matter (should be combination).
• Computing factorials of very large numbers without simplification.
• Not verifying that r ≤ n (you cannot select more elements than available).

Pro tip

The key question is: does order matter? If yes, use permutations. If no, use combinations. For example, in a lottery order does not matter (combination), but in a horse race it does (permutation).