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MMathematicsICTS4 to S6Counting

Counting lab: passwords, permutations and combinations

Is your phone PIN safe? Count every possible password and discover why longer passwords are so much stronger.

▶ Start the lab🧠 Questions📄 My report
🔐kⁿ · ⁿPᵣ · ⁿCᵣcounting, permutations and combinations
LevelS4 to S6 (and A Level)
TimeAbout 35 min
You usekⁿ, n!, ⁿPᵣ, ⁿCᵣ
PriceFree: no account

🎯 Aim

To use the counting principle, permutations and combinations.

You will learn to

  • Use the multiplication (counting) principle.
  • Calculate n!, permutations and combinations.
  • Decide when order matters.
  • Explain password strength.

🧭 What to do

  1. Choose the Characters allowed and change the Length.
  2. Read the number of passwords and the time to try them all.
  3. Change n and r and compare ⁿPᵣ and ⁿCᵣ.
  4. Explain why ⁿPᵣ is bigger.
  5. Answer the questions with the class data.
  6. Write your conclusion and download your lab report.

🔬 Lab bench

📋 The data

Numbers of possible passwords.

🧠 Investigate: answer from the data

Type your answer and tap Check. Use 💡 Hint first; open the method only when you are stuck.

📄 My lab report

Your report collects the aim, your data table, the graphs, your answers and your conclusion. It is made on your phone or computer: nothing is sent to anyone.

Check your conclusion

With k characters and length n there are kⁿ passwords, so each extra character multiplies the number by k: length matters most. Permutations count ordered choices, ⁿPᵣ = n!/(n − r)!; combinations ignore order, ⁿCᵣ = n!/(r!(n − r)!), so ⁿPᵣ = r! × ⁿCᵣ.

➕ Get more from this experiment

📝 Free

Print the worksheet (button above), download the data, and save your report. Teachers can project this page in class.

📦 Premium Lab Pack

1 000 RWF

  • Worksheet: counting problems with PINs, number plates and teams
  • Permutations and combinations problems
  • Teacher guide and answer key
  • Exam-style questions with marking scheme
See the pack

👩‍🏫 For teachers

Use it as a practical when apparatus is missing, as homework, or as a project. Free lesson plans help you plan, teach and assess.

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