How many ways can you pick 2 people out of 5 — and does it change if they end up in specific seats? The first hurdle in any counting problem is deciding between permutations (order matters) and combinations (order does not). Pick the wrong one and every later step is off.
This article gives you a single question that settles the choice, works through the formulas by hand, and computes the odds of Loto 6 — then verifies everything with a free browser tool.
The difference in one question
Ask yourself:
After choosing, do I also count the ways to arrange what I chose?
If yes, use a permutation. If no, use a combination.
| Aspect | Permutation (nPr) | Combination (nCr) |
|---|---|---|
| What it counts | Ways to choose r from n and arrange them | Ways to choose r from n, unordered |
| Order | Distinct (A,B and B,A differ) | Ignored (A,B and B,A are the same) |
| Formula | nPr = n! / (n − r)! | nCr = n! / (r! × (n − r)!) |
| Example (2 of 5 people) | 5P2 = 20 | 5C2 = 10 |
Notice the extra r! in the combination denominator. It divides away the orderings that a combination should not distinguish. That observation also gives the exact relationship between the two:
nPr = nCr × r!
A permutation is simply a combination multiplied by the number of arrangements. Once you have either value, the other follows.
Some everyday practice with that one question:
| Situation | Judgment | Count |
|---|---|---|
| Choose 1 class representative from 10 | Order irrelevant | 10C1 = 10 |
| Seat 5 people in a row | Every order is a different outcome | 5P5 = 5! = 120 |
| Draw 5 cards from 52 | Hand order irrelevant | 52C5 = 2,598,960 |
| Pick 1st, 2nd and 3rd place from 10 | Swapping ranks changes the result | 10P3 = 720 |
Factorial notation: n! means n × (n−1) × ⋯ × 1, with the convention 0! = 1. Reading it as "there is one way to arrange nothing" keeps the formulas consistent when n = r or r = 0.
Worked examples
Example 1: permutation — arrange 2 of 5 people
The first seat has 5 candidates and the second has the remaining 4, so 5P2 = 5 × 4 = 20. The formula agrees: 5! / (5 − 2)! = 120 / 6 = 20.
Example 2: combination — choose 2 of 5 people
The 20 arrangements above include pairs like (A,B) and (B,A). Each unordered pair was counted 2! = 2 times, so 5C2 = 20 ÷ 2 = 10. By formula: 5! / (2! × 3!) = 120 / (2 × 6) = 10. The relation checks too: 5P2 = 5C2 × 2! = 10 × 2 = 20.
Example 3: the odds of Loto 6
In Loto 6 you pick 6 numbers out of 43, and the first prize requires matching all six main numbers (the official rules are published by Mizuho Bank, which operates the lottery site). Since only the set of numbers matters, this is a combination:
43C6 = 6,096,454
A single entry therefore wins first prize with probability 1 in 6,096,454. For contrast, 43P6 gives 6,096,454 × 720 — about 4.39 billion — because each set would be tallied once per ordering. The draw ignores order, so combination is right.
A two-step trick: build a permutation from a combination
Splitting “choose, then arrange” into stages keeps problems readable. To pick 3 of 10 people and rank them 1st–3rd: first count the selections, 10C3 = 120, then multiply by the arrangements, 3! = 6, for 10P3 = 120 × 6 = 720. Each stage has a clear meaning, which makes mistakes easier to spot.
Three common mistakes: using a combination when order carries meaning (seating charts, rankings); entering n < r (you cannot choose 7 of 5 — the formula breaks down); and applying nCr when repeats are allowed (selections with repetition need separate formulas, which this tool does not cover). Checking the condition n ≥ r ≥ 0 with distinguishable items prevents all three.
Verify your answers with the tool
For checking hand calculations and handling huge numbers, Tools Hub’s Permutation & Combination Calculator computes nCr, nPr and n! with BigInt arbitrary-precision integers — exact to about 150 digits — and shows the substituted working for every result.
How to use it (3 steps)
Pick the calculation type
Choose one of the three tabs: Combination (nCr), Permutation (nPr) or Factorial (n!). Selecting only? Use nCr. Arranging? Use nPr.
Enter n and r
Type the total count n and the number chosen, r. For the lottery example, the 43C6 preset button fills both in.
Read the result and the working
The result card shows the total (for example 6,096,454) and the Working panel below shows the substituted formula. The copy button outputs a line like C(43, 6) = 6,096,454.
The tool featured in this article
Permutation & Combination Calculator
Compute nCr, nPr and factorials with the working shown. Exact BigInt arithmetic, a Loto 6 preset, free and browser-based.
To try the article’s example, switch the tool to Combination (nCr), enter n = 5 and r = 2, and read 10 with the substituted working below. Flipping the tab to Permutation (nPr) turns the same input into 20.
For the fraction reduction inside the formulas, see the guide to working with fractions; for putting counts to work on data, see mean, median and standard deviation.
Summary
- Permutations count arrangements (nPr = n!/(n−r)!); combinations count selections (nCr = n!/(r!(n−r)!))
- One question decides: after choosing, does order still matter?
- The two connect through nPr = nCr × r!, so either value converts into the other
- Loto 6 first-prize odds follow from the official rule (match all six of six numbers) via 43C6 = 6,096,454
- Verify hand calculations with a tool that shows its working
FAQ
Why is 0! equal to 1?
Because there is exactly one way to arrange nothing. With 0! = 1 the formulas stay consistent: nPr at n = r gives n! / 0! = n!, and nC0 = nCn = 1 throughout.
Why is Loto 6 a combination rather than a permutation?
Matching the draw depends only on which numbers you hold, not the order on your entry slip. Counting with 43P6 would count each set of numbers 720 times.
When should I use nPr instead of nCr?
Ask whether swapping two chosen items produces a different outcome. Seats, rankings and codes do, so they are permutations. Committees and hands do not, so they are combinations. In word problems, the final step telling you: arrange means nPr, just select means nCr.
What about nC1 and nCn?
nC1 = n (one item can be chosen n ways) and nCn = 1 (one way to take everything). The symmetry nCr = nC(n−r) — for instance 5C2 = 5C3 = 10 — also helps when r is close to n.
How large can the numbers get?
The tool accepts n and r up to 500. Results reach about 150 digits, and BigInt arithmetic keeps them exact — far beyond what an ordinary calculator displays before overflowing.
References
- OpenStax, College Algebra 2e, Section 9.5 “Counting Principles” (permutation and combination formulas): openstax.org
- Mizuho Bank, Loto 6 (official winning conditions and prize tiers): www.mizuhobank.co.jp
