BIZOMAT MAKEOS · Mathematics
Factorials, Permutations and Combinations
How many arrangements or selections of r items from n are possible?
Factorial n!, permutations nPr and combinations nCr, with large values in scientific notation, for sampling plans, tooling options and probability.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Total items n | items | Number of distinct items available (up to 170). |
| Items chosen r | items | Number of items chosen. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Total items n
- What it is
- How many different items you have to choose from, for example the number of parts, people or positions available.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 170 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- n! and both counts grow very quickly with n.
- Get it right
- The items must be distinct; repeated identical items need a different counting method.
What this means: Items chosen r
- What it is
- How many of the n items are picked at a time. Permutations count each order separately; combinations count each group once regardless of order.
- Units and format
- Enter a number in items: a count of items. The form accepts 0 items to 170 items. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 items as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- r cannot be larger than n.
Method and formulas
- n! = 1 × 2 × … × n (0! = 1).
- nPr = n! ÷ (n − r)!.
- nCr = n! ÷ (r! (n − r)!).
- Example: from 10 items, 3 can be arranged in 720 ways and chosen in 120 ways.
Limitations
- Values beyond about 10¹⁵ are shown in floating point and are approximate in the last digits.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Count possible tool or set-up combinations
- Support sampling and probability calculations
- Check combinatorics in test plans
Who it is for
- Engineers
- Technicians
- Engineering students
What you get
- n!
- Permutations nPr (order matters)
- Combinations nCr (order does not matter)
Interpreting the result
- Use combinations when order does not matter, permutations when it does.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 19. Topic reference; no source tables or recommended settings are reproduced.
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