Combination Calculator
nCr and nPr counts plus 1 in odds for any pick.
What it does
Combinations count selections where order means nothing (lottery lines, committees), while permutations count arrangements where order matters (podiums, passwords). Same pool and pick, wildly different counts.
The tool above takes pool n with pick r. It returns both counts with the one ticket odds beside them.
The default 49 choose 6 gives 13,983,816 combinations against 10,068,347,520 permutations: order multiplies the count 720 fold (6!). The same pair powers classic lottery math.
Formulas follow Khan Academy counting units with the same factorial ratios in LibreTexts math references. Both cap exact integer work where doubles hold every digit.
How to use
- Enter the pool size n, 1 to 100.
- Enter the pick count r from 0 to n.
- Read combinations first for unordered picks.
- Read permutations for ordered arrangements.
- Use the odds line for one exact ticket chances.
Formula
- nCr
- Order ignored: lottery lines, teams.
- nPr
- Order counted: podiums, codes.
- r!
- Divider removing the orderings.
Sums compute multiplicatively to stay exact inside double range. r past n is rejected; r zero reads one way (choose nothing).
Worked example
Example: 49 choose 6
Pool 49, pick 6.
- Combinations: 49! / (6! x 43!) = 13,983,816.
- Permutations: 49 x 48 x 47 x 46 x 45 x 44 = 10,068,347,520.
- Ratio: 10,068,347,520 / 13,983,816 = 720 = 6!.
- Odds: one exact line wins 1 in 13,983,816.
- Tool shows all three.
Reference table
| Pick | nCr | nPr |
|---|---|---|
| 10 choose 2 | 45 | 90 |
| 10 choose 7 | 120 | 604,800 |
| 20 choose 3 | 1,140 | 6,840 |
| 49 choose 6 | 13,983,816 | 10,068,347,520 |
Limits
- No repeats assumed; replacement needs powers instead.
- Pool caps at 100 for exactness.
- Distinct items only; duplicates divide counts.
- Huge pools overflow display past double range.
Common mistakes
- Using permutations for lottery lines (720x too many).
- Allowing r past n.
- Forgetting order matters in codes and podiums.
- Reading odds as percent directly.
- Trusting rounded counts for exact tickets.
Try it yourself
Quick checks. Answers sit tucked under each question.
10 choose 7?
120 combinations, 604,800 permutations.
20 choose 3?
1140 combinations, 6840 permutations.
5 choose 5?
1 combination; order of all five still 120 ways.
Questions
Combinations vs permutations?
Combinations ignore order (13.9M for 49/6); permutations count it (10B).
What is 49 choose 6?
13,983,816: the classic 6/49 lottery line count.
When does order matter?
Podiums, passwords, PINs. Lottery lines and teams ignore it.
What is nCr when r is 0?
1: exactly one way to choose nothing.
Why cap at 100?
Exact integer display ends where doubles lose digits.
How do odds follow?
One exact ticket wins 1 in the combinations count.