What this tool does
Calculate combinations (n choose r), the number of ways to pick r items from n items when the order of picks does not matter. You fill in these fields: Total items (n) and Items to choose (r). It shows combinations C(n, r).
Inputs
- Total items (n): A whole number. Must be at least 0.
- Items to choose (r): A whole number. Must be at least 0.
How to use
- Enter the total items (n).
- Enter the items to choose (r).
- Read the result: combinations C(n, r). It appears once every required field has a valid value.
Worked examples
Example 1. Inputs:
- Total items (n): 52
- Items to choose (r): 5
Results:
- Combinations C(n, r): 2,598,960
How this is worked out: C(52,5) = 52 x 51 x 50 x 49 x 48 / (5 x 4 x 3 x 2 x 1) = 311875200 / 120 = 2598960. This is the number of five-card poker hands from a 52-card deck.
Example 2. Inputs:
- Total items (n): 15
- Items to choose (r): 5
Results:
- Combinations C(n, r): 3,003
How this is worked out: C(15,5) = 15 x 14 x 13 x 12 x 11 / 120 = 360360 / 120 = 3003.
Example 3. Inputs:
- Total items (n): 50
- Items to choose (r): 6
Results:
- Combinations C(n, r): 15,890,700
How this is worked out: C(50,6) = 50 x 49 x 48 x 47 x 46 x 45 / 720 = 11441304000 / 720 = 15890700. Checked: 720 x 15890700 = 11441304000.
Example 4. Inputs:
- Total items (n): 7
- Items to choose (r): 0
Results:
- Combinations C(n, r): 1
How this is worked out: Choosing none of the items is one way. C(n,0) = 1 for every n.
Limitations
- Counts only the number of ways to choose. It does not list the actual groups.
- Treats every item as different. It does not handle lists where some items are identical.
- Results above about 9 quadrillion may not be exact, because they are larger than a browser can hold as a whole number.
FAQs
- Q: What is the difference between a combination and a permutation?
- A: A combination ignores order, so picking Ana then Ben is the same pick as Ben then Ana. A permutation counts order as well. This tool counts combinations only.
- Q: Why does the tool show an error when r is bigger than n?
- A: You cannot choose more items than the group holds, so there is no valid answer to show. Make r equal to or smaller than n.
- Q: Can I enter decimals?
- A: No. Both n and r must be whole numbers, because you cannot pick part of an item.
- Q: What happens when the answer is too large to show?
- A: The tool stops with a message that the result is too large. It does not show a wrong number in its place.
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