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Learn through experiments

Count only after naming the outcomes

How do order and reuse change the number of possibilities?

Entering midway? Multiply choices across positions; divide only when each unordered outcome was counted the same number of times.

1 · Try

Start with a concrete question

Choose k symbols from n distinct symbols under three explicit rules. The shared k control is capped at n.

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

2 · Predict

What do you expect?

Predict before calculating: How do order and reuse change the number of possibilities?

3 · Experiment

Change one thing. Inspect what happens.

  1. List outcomes for n=3, k=2 under all three rules.
  2. Explain why AB and BA coincide only in the unordered case.
  3. Set k=0 and interpret the single empty outcome.
  4. Create a counting scenario and justify its rule before calculating.
Experiment inputs
Inputs
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Requested positions2 positions
Effective positions2 positions
Possible outcomes16 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.
016
  • Ordered with repetition (outcomes)16
    Ordered with repetition (outcomes): 16Horizontal bar from zero to 16, on the shared zero to 16 numeric axis.
  • Ordered without repetition (outcomes)12
    Ordered without repetition (outcomes): 12Horizontal bar from zero to 12, on the shared zero to 16 numeric axis.
  • Unordered without repetition (outcomes)6
    Unordered without repetition (outcomes): 6Horizontal bar from zero to 6, on the shared zero to 16 numeric axis.

Follow the calculation

  1. Available distinct symbols n=4; requested k is capped to 2≤n for this shared comparison.
  2. Ordered choices with repetition: 4 choices at each of 2 positions → 4^2=16.
  3. Choosing zero items has one outcome: the empty selection.
  4. The comparison bars use the same n=4 and effective k=2 for all three rules. Only the rule changes; all bars count outcomes.

Before counting, specify whether order matters and whether reuse is permitted. The same objects give different sample spaces under different rules.

Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.

4 · Understand

Explain the mechanism

Count only after naming the outcomes

Choose k symbols from n distinct symbols under three explicit rules. The shared k control is capped at n.

Concepts to recall

Multiply choices across positions; divide only when each unordered outcome was counted the same number of times.

Understand the mechanism

With four symbols and two positions, repetition gives 4²=16 ordered strings.

Without repetition there are 4×3=12 ordered strings.

Unordered pairs count each pair twice, so 12/2!=6. In general, divide by k! because all k! permutations describe the same set.

Symbols and calculation

With repetition n^k; ordered distinct n!/(n−k)!; unordered distinct C(n,k)=n!/[k!(n−k)!]. Factorial n! multiplies 1 through n; 0!=1.

Worked example

Available distinct symbols n=4; requested k is capped to 2≤n for this shared comparison.

Ordered choices with repetition: 4 choices at each of 2 positions → 4^2=16.

Choosing zero items has one outcome: the empty selection.

The comparison bars use the same n=4 and effective k=2 for all three rules. Only the rule changes; all bars count outcomes.

Your investigation

  1. List outcomes for n=3, k=2 under all three rules.

  2. Explain why AB and BA coincide only in the unordered case.

  3. Set k=0 and interpret the single empty outcome.

  4. Create a counting scenario and justify its rule before calculating.

Limits and interpretation

  • These are bounded teaching models with explicit assumptions, not evidence of a real-world intervention.
5 · Apply

Try a fresh case

How many unordered pairs can be selected from five distinct objects?

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

Explain a fresh small numerical case with a diagram or table and an argument.

College investigation

Implement the method, test a boundary or counterexample, and justify its assumptions.

Workplace investigation

Use a small decision scenario, state units and constraints, and test how the recommendation changes.

6 · Continue

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.