Understanding Permutations and Combinations
Permutations and combinations are fundamental tools for solving counting problems: determining how many ways to arrange items, select teams, distribute prizes, or organize data. The difference between them is crucial: permutations count ordered arrangements (where sequence matters), while combinations count unordered selections (where sequence doesn't matter). For example, "how many 3-digit passwords can I make from 5 digits?" (permutation) is different from "how many 3-person committees can I form from 5 people?" (combination). These calculators solve these problems using mathematical formulas that would be tedious to compute by hand for large numbers.
The Fundamental Counting Principle
The Multiplication Principle: If one task can be done in m ways and a second task can be done in n ways, both tasks can be done in m × n ways. If you're making a sandwich with 5 bread choices and 7 filling choices, there are 5 × 7 = 35 different sandwiches. This principle underlies all permutation and combination calculations.
Factorial (n!): The factorial of n is the product of all positive integers from 1 to n. 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials grow extremely quickly: 10! = 3,628,800, and 20! ≈ 2.4 trillion. By definition, 0! = 1. Factorials are the foundation of permutation and combination formulas.
Permutations: Order Matters
A permutation is an ordered arrangement of items. The number of permutations of n items taken r at a time (denoted nPr or P(n, r)) counts all ways to select and arrange r items from n distinct items, where order is important.
This formula says: take n! (all arrangements of n items), then divide by (n-r)! (to remove the arrangements of items we didn't select). For example, nP3 when n = 5 = 5! ÷ (5-3)! = 120 ÷ 2 = 60. There are 60 ways to select and arrange 3 items from 5.
Combinations: Order Doesn't Matter
A combination is an unordered selection. The number of combinations of n items taken r at a time (denoted nCr or C(n, r)) counts all ways to select r items from n distinct items, where order is unimportant.
This formula divides nPr by r! to account for the fact that different orderings of the same r items are counted as one combination. For example, nC3 when n = 5 = 120 ÷ (6 × 2) = 120 ÷ 12 = 10. There are 10 ways to select 3 items from 5 (ignoring order).
Worked Example: Lottery and Committee Selection
| Problem |
Type |
Formula |
Result |
| How many 4-digit PINs can be made from 10 digits (0-9)? |
Permutation (order matters, repetition allowed) |
10 × 10 × 10 × 10 |
10,000 |
| How many ways can 3 gold, 2 silver, 1 bronze medals be awarded to 6 athletes? |
Permutation (order matters, no repetition) |
6P3 = 6!/(6-3)! = 120 |
120 ways |
| How many 6-number lottery combinations from 1-49? |
Combination (order doesn't matter, no repetition) |
49C6 = 49!/(6! × 43!) |
10,068,347 |
| How many 5-person committees can be formed from 15 people? |
Combination (order doesn't matter) |
15C5 = 15!/(5! × 10!) |
3,003 |
Notice that 6P3 = 120 (permutations of 3 from 6) is much larger than if we calculated nC3. The difference is that permutations count every possible ordering, while combinations count unordered groups.
When to Use Permutations vs. Combinations
Use Permutation (nPr) when: You're arranging items in a specific order (seating arrangements, race placements, password creation, phone number arrangements). The question often includes words like "arrange," "order," "sequence," or "how many ways can we line up?"
Use Combination (nCr) when: You're selecting items where order doesn't matter (committee selection, pizza toppings, lottery numbers, choosing team members). The question often includes words like "select," "choose," "committee," or "how many ways to pick?"
Limitations and Assumptions
- No Repetition Assumed: Formulas assume items are distinct and cannot be selected twice. If repetition is allowed, different formulas apply.
- Distinct Items: All n items are considered different. If some items are identical, results overcount.
- Large Numbers: Factorials grow extremely fast. Calculations above 10,000! may lose precision due to computational limits.
- r ≤ n Required: You cannot select more items than available. If r > n, the result is 0.
- No Partial Selections: These formulas count complete arrangements/selections. Partial selections follow different logic.
When to Use This Calculator
Use this calculator to solve any counting problem that requires nPr or nCr. Determine the number of possible outcomes for probability calculations. Verify your hand calculations for combinatorics homework. Solve real-world problems: lottery odds, committee selections, seating arrangements, password possibilities, or any scenario requiring systematic counting. Remember: permutations give larger numbers than combinations because they count order as distinct.