Count teams and ordered roles
When does order create a different outcome?
Entering midway? Multiply choices for successive positions when their eligible counts are known. Factorial n! multiplies whole numbers from 1 through n.
Start with a concrete question
Choose distinct people from a small fictional group. Compare assigning ordered roles with forming an unranked team.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict how many times the same three-person team is counted among ordered role assignments.
Change one thing. Inspect what happens.
- Count captain, deputy and recorder assignments among six people.
- List the orders of one three-person team and divide out the repetition.
- Request more people than available and inspect the explicit effective-size cap.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Compare the current results
Every bar starts at zero and uses the same numeric scale in this chart. The upper bound automatically adjusts to the largest current value (at least 1), so compare the numbers when changing inputs. Units and quantities come from each label; different kinds of quantity are not interchangeable. Displayed values are rounded to at most six significant digits.Follow the calculation
- Select 3 distinct people from 6.
- Ordered positions: 6 × 5 × 4 = 120.
- Each team appears in 3! = 6 orders. Divide 120/6 = 20.
Count outcomes according to the question: a captain and deputy are ordered roles, but an unranked team is not.
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Explain the mechanism
Count teams and ordered roles
Recap for entering here
Multiply choices for successive positions when their eligible counts are known. Factorial n! multiplies whole numbers from 1 through n.
The concrete question
Choose distinct people from a small fictional group. Compare assigning ordered roles with forming an unranked team.
Predict how many times the same three-person team is counted among ordered role assignments.
Work through the mechanism
Ordered assignments without replacement have shrinking choices: 6 choices, then 5, then 4, for 120 assignments.
An unranked team appears 3!=6 times among those assignments. Dividing 120 by 6 gives 20 teams.
A probability based on counts requires equally likely elementary outcomes. A team-selection procedure that favors some people cannot use the uniform fraction without justification.
Formula and symbols
Ordered count = n!/(n−k)!; unordered count C(n,k)=n!/[k!(n−k)!], for 0≤k≤n.
Test the idea
Count captain, deputy and recorder assignments among six people.
List the orders of one three-person team and divide out the repetition.
Request more people than available and inspect the explicit effective-size cap.
Assumptions and boundaries
Selection is without replacement and all people are distinct.
Uniform random selection is not simulated; counts alone do not establish real-world probabilities.
Extend your investigation
High school: Enumerate teams from four names and compare with the formula.
College: Derive binomial coefficients and relate them to repeated binary trials.
Working professionals: Audit whether a selection or assignment policy makes outcomes equally likely.
Try a fresh case
How many unordered two-person teams can be selected from five people?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Enumerate teams from four names and compare with the formula.
College investigation
Derive binomial coefficients and relate them to repeated binary trials.
Workplace investigation
Audit whether a selection or assignment policy makes outcomes equally likely.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.