Study Guide
Probability: Tree Diagrams and Conditional Probability
A senior study guide to multi-stage probability: the addition and multiplication rules, tree diagrams with and without replacement,…
- Difficulty
- Level 2 · Standard
- Time
- 25 min
WorksheetYear 12Level 2 · Standard
Free Year 12 Counting Techniques: Permutations and Combinations Worksheet
A senior maths worksheet on counting: the multiplication principle, factorials, permutations (including repeated letters, restrictions and circular arrangements) and combinations, then using counting to find probabilities. Each section opens with a short summary and worked example, and the sheet finishes with an extended committee problem. Full working is given for every answer.
Instant download · No sign-up · Free for personal, classroom and homeschool use
- Year
- Year 12
- Subject
- Maths
- Topic
- Probability
- Difficulty
- Level 2 · Standard
- Estimated time
- 45 minutes
- Curriculum
- New Zealand Curriculum
- Questions
- 17
- Answers
- Included, with working
- Format
- PDF (A4), print or complete online
Students will practise
- using the multiplication principle and factorial notation to count arrangements
- calculating permutations with nPr, including repeated objects and restrictions
- calculating combinations with nCr and deciding whether order matters
- finding probabilities by counting favourable and total selections
Curriculum: New Zealand Curriculum. We show specific outcome codes only where they have been verified against the official curriculum document.
What's next?
Completed: Counting Techniques: Permutations and Combinations
How to use this worksheet
- Print the PDF or complete it on screen. Write the student's name and date at the top.
- Read the instructions for each part together before starting. Encourage working to be shown, not just answers.
- Mark with the answer sheet. Compare the method, not only the final answer, and give credit for correct working.
- Record the score, celebrate what went well, then follow the "What's next" suggestion.
Common questions
Who is this worksheet for?
Year 12 students (typically ages 16–17) working on probability. It is pitched at level 2 · standard; a harder version is linked above.
Are the answers included?
Yes. The PDF includes a full answer sheet with the working for every question. You can also view the answers online.
How long does it take?
About 45 minutes for 17 questions. Short, regular sessions work best: two or three a week beats one long one.
Do I need to sign up to download?
No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.
What should we do next?
Try Discrete Random Variables and Expected Value.
Preview
Success TutoringWorksheetsYear 12 · Mathematics · ProbabilityCounting Techniques: Permutations and Combinations
Name
Date
Score /
Show your working. Leave answers as whole numbers or fractions in simplest form unless a decimal is asked for.
Part A: The multiplication principle and factorials
Multiplication principle: if one choice can be made in m ways and a second choice in n ways, the two together can be made in m × n ways. Draw a box for each position and write the number of choices in it.
Factorials: n! = n × (n − 1) × … × 2 × 1 is the number of ways to arrange n different objects in a row. By definition 0! = 1.
Worked example. How many 3-digit numbers can be made from the digits 1 to 6 if no digit is repeated? Boxes: 6 × 5 × 4 = 120. If repeats were allowed it would be 6 × 6 × 6 = 216.
1.A PIN is made of 4 digits, each from 0 to 9, and digits may repeat. How many different PINs are possible? (answer space)
2.In how many different orders can 5 different books be placed on a shelf? (answer space)
3.Without a calculator, = (answer space).
4.A code is made of 3 different letters chosen from the 26 letters of the alphabet (no letter repeated). How many codes are possible?
- A17 576
- B15 600
- C2600
- D78
Part B: Permutations (order matters)
A permutation is an arrangement where order matters. The number of ways to arrange r objects chosen from n different objects is [sup n]P[sup r] = [frac n!/(n − r)!].
If some objects are identical, divide by the factorial of each repeated group: the letters of a word with n letters, where one letter appears a times and another b times, can be arranged in ways. For objects that must be together, glue them into one block, arrange the blocks, then arrange inside the block.
Worked example. In how many ways can a president and a secretary be chosen from 10 club members? Order matters (the roles differ): 10P2 = 10 × 9 = 90.
5.8 runners compete in a race. In how many ways can the gold, silver and bronze medals be awarded?
6.How many different arrangements are there of the letters of the word PARALLEL?
7.6 friends sit in a row of 6 seats. In how many ways can they sit if Priya and Sam must sit next to each other?
8.There are 5! = 120 different ways to seat 5 people around a circular table (where only the order around the table matters). ☐ True ☐ False
Part C: Combinations (order does not matter)
A combination is a selection where order does not matter. The number of ways to choose r objects from n different objects is [sup n]C[sup r] = [frac n!/r! (n − r)!]. Each combination of r objects can be arranged in r! orders, so nCr = .
Ask yourself: if I swap two of the chosen objects, do I get a different outcome? If yes, use permutations; if no, use combinations. For at least one questions, count everything and subtract the selections with none.
Worked example. How many ways can a team of 3 be chosen from 8 players? 8C3 = = = 56. Compare with 8P3 = 336: each team is counted 3! = 6 times when order matters.
9.Evaluate 10C3. (answer space)
10.A committee of 4 is chosen from 7 women and 5 men. How many committees contain exactly 2 women and 2 men?
11.Using the same 7 women and 5 men, how many committees of 4 contain at least one man?
12.Which situation should be counted with combinations rather than permutations?
- AChoosing 3 students from a class to form a cleaning team
- BAwarding first, second and third prizes in an art show
- CMaking a 4-digit code with no repeated digits
- DArranging 6 photos in a row on a wall
Preview — the PDF contains the full resource and answer sheet.
More Year 12 probability resources
- Worksheet
Discrete Random Variables and Expected Value
A senior maths worksheet on discrete probability distributions: checking that a table is a valid distribution, finding unknown…
- Difficulty
- Level 3 · Challenge
- Time
- 45 min
- Questions
- 15 questions
- Answers
- Answers included
Probability in other year levels
- Year 10 Probability Worksheet — Level 1
Worksheet · Year 10 · Probability · Level 1 · Easier
- Year 10 Probability Worksheet — Level 2
Worksheet · Year 10 · Probability · Level 2 · Standard
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About this resource
- Created by
- Success Tutoring
- Last reviewed
- 1 October 2026
- How it was made
- Written by our team
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