OC counting combinations: 8 problems with lists and tree diagrams

Quick Answer

To count every case without guessing, fix the first choice, write every option that can follow it, then move to the next first choice. An organised list or a tree shows each case once, and multiplying the number of choices at each step gives the same total. This page from Braintree Coaching Australia has eight original problems for Year 3–4 children preparing for the NSW Opportunity Class (OC) test, two worked models and a reason for every wrong option. It is free and needs no sign-up.

Who is this counting practice for?

This set suits a Year 3 or Year 4 child who can add and multiply small numbers and is ready to ask "how many ways?" It is skill practice for careful, organised thinking. We do not claim that questions like these appear on the real test, and your child does not need any special word for what they are doing.

The NSW Department of Education describes the Opportunity Class Placement Test as a computer-based test with three sections, and all questions are multiple choice. The department's own OC practice tests are the place to see the real format.

How do we count every case without guessing? Two worked models

Both models solve the same problem. A café lets you pick one drink (milk or juice) and one snack (apple, muffin or crackers). How many different drink-and-snack choices are there?

Model 1: an organised list

Fix the first choice, then list every snack that can follow it.

1. Milk with apple
2. Milk with muffin
3. Milk with crackers
4. Juice with apple
5. Juice with muffin
6. Juice with crackers

There are 6 choices. The list is organised: all the milk lines come first, then all the juice lines. If a line is missing, the gap is easy to see.

Model 2: a tree

A tree shows the same cases as branches. Each drink is a branch, and each snack is a smaller branch that grows from it.

Start
├── Milk
│   ├── apple
│   ├── muffin
│   └── crackers
└── Juice
    ├── apple
    ├── muffin
    └── crackers

Text equivalent of the tree. There are two first branches, Milk and Juice. Each first branch has three second branches: apple, muffin and crackers. The tree ends in 3 + 3 = 6 final branches, so there are 6 choices. This is the same count as the list.

Multiplying gives the same answer: 2 drinks × 3 snacks = 6. Use multiplying to check a list, not to replace it. When a problem has a rule or when order does not matter, a list shows what the multiplication would miss.

The questions

Work on paper. List or draw a tree for every problem first, then check by multiplying where it fits.

Question 1. A school canteen has 3 mains (pasta, rice bowl or wrap) and 2 drinks (milk or juice). A child picks one main and one drink. How many different choices are there?

• A. 6
• B. 4
• C. 5
• D. 12

Question 2. A lunch box has one sandwich and one fruit. There are 5 sandwiches and 3 fruits to choose from. How many different lunch boxes can be made?

• A. 8
• B. 13
• C. 15
• D. 30

Question 3. Mia picks one shirt (red, white or green), one pair of shorts (black or navy) and one hat (cap or bucket). How many different outfits can she make?

• A. 7
• B. 10
• C. 6
• D. 12

Question 4. Sam picks one top from 3 tops, one skirt from 2 skirts and one pair of socks from 3 pairs. How many different outfits can Sam make?

• A. 8
• B. 18
• C. 15
• D. 6

Question 5. A padlock code has 2 digits. Each digit can be 1, 2, 3 or 4, and the same digit can be used twice (for example 3 then 3). How many different codes are there?

• A. 16
• B. 8
• C. 12
• D. 14

Question 6. A different padlock code has 2 digits from 1, 2, 3 and 4, but a digit cannot be used twice. The code 1 then 3 is different from 3 then 1. How many different codes are there?

• A. 16
• B. 8
• C. 6
• D. 12

Question 7. A café offers 3 mains (pasta, fish or chicken) and 4 sides (rice, chips, salad or beans). A meal is one main and one side. The fish is never served with chips. How many different meals can be ordered?

• A. 12
• B. 7
• C. 11
• D. 10

Question 8. A teacher chooses 2 children from Asha, Ben, Chi and Dev to carry the flag. The two children are chosen together, so choosing Asha and Ben is the same as choosing Ben and Asha. How many different pairs can be chosen?

• A. 12
• B. 6
• C. 5
• D. 7

Answers and worked explanations

Check one question at a time. For each wrong answer, find the option your child chose and read why it is tempting.

Question 1 answer: A, 6

List by main. Pasta with milk, pasta with juice, rice bowl with milk, rice bowl with juice, wrap with milk, wrap with juice. That is 6. Option B (4) is a list that missed two choices. Option C (5) adds 3 + 2 instead of pairing each main with each drink. Option D (12) counts "pasta with milk" and "milk with pasta" as two choices, but they are the same meal.

Question 2 answer: C, 15

Take sandwich 1 and write it with fruit 1, 2 and 3. That is 3 lunch boxes. Do the same for sandwiches 2, 3, 4 and 5, which gives five groups of 3, so 15. Option A (8) adds 5 + 3. Option B (13) is a list that missed two lunch boxes. Option D (30) counts each sandwich-and-fruit pair twice, once in each order.

Question 3 answer: D, 12

Fix the shirt and the shorts first. Red with black, red with navy, white with black, white with navy, green with black and green with navy make 6 pairs. Each pair can have the cap or the bucket hat, so 6 × 2 = 12. Option A (7) adds 3 + 2 + 2. Option B (10) is a list with two outfits missing. Option C (6) stops after the shirt and shorts and forgets the hat.

Question 4 answer: B, 18

Three tops and 2 skirts give 6 top-and-skirt pairs. Each pair can have any of the 3 pairs of socks, so 6 × 3 = 18. Option A (8) adds 3 + 2 + 3. Option C (15) is an unorganised list that lost a group of 3. Option D (6) uses only the top and skirt and forgets the socks.

Question 5 answer: A, 16

Write every code that starts with 1: 11, 12, 13, 14. The same four endings follow 2, 3 and 4. That is 4 groups of 4, so 16. Repeats such as 11 and 22 are allowed. Option B (8) adds 4 + 4. Option C (12) leaves out the four repeat codes, but the question says the same digit can be used twice. Option D (14) is a list with two codes missing.

Question 6 answer: D, 12

Write every code that starts with 1 but does not repeat the 1: 12, 13, 14. That is 3. The same applies to 2, 3 and 4, so 4 groups of 3 give 12. Option A (16) ignores the rule and counts codes like 11 and 22. Option B (8) adds 4 + 4. Option C (6) treats 13 and 31 as one code, but the question says they are different.

Question 7 answer: C, 11

Without the rule there are 3 × 4 = 12 meals. List them by main. Pasta has 4 sides. Chicken has 4 sides. Fish has rice, salad and beans only, which is 3 sides. That is 4 + 4 + 3 = 11. Option A (12) ignores the rule. Option B (7) adds 3 + 4. Option D (10) takes away too much, treating the fish rule as if it removed two meals.

Question 8 answer: B, 6

Fix the first child and pair them with the children who come later in the list. Asha with Ben, Asha with Chi, Asha with Dev, Ben with Chi, Ben with Dev, Chi with Dev. That is 6 pairs, and every pair appears once. Option A (12) counts both Asha with Ben and Ben with Asha. Option C (5) is a list that dropped one pair. Option D (7) adds 4 + 3.

What mistake patterns should we look for?

Tally which kind of error came up. Each one needs a different fix.

Pattern	Seen in	What to practise next
Added the choices instead of pairing them	Questions 1 to 8	Write the first two or three cases out in full before any arithmetic
Short count from an unorganised list	Questions 1 to 5 and 8	List by first choice, one group at a time, and count each group
Counted the same pair twice, or treated different orders as one	Questions 1, 2, 6 and 8	Say whether order matters in the question, then fix a listing rule
Ignored a rule in the question	Questions 5, 6 and 7	Underline the rule, then cross out the cases it removes

Can my child build their own tree?

Ask your child to invent a problem with 3 choices for the first step and 2 for the second, for example 3 hats and 2 scarves. They draw the tree, write the list beside it and say the total.

Self-check: the tree has 3 first branches, each with 2 second branches, so 6 final branches. The list should have 6 lines in 3 groups of 2. If the tree and the list disagree, find the group that has a missing or extra line.

What should we do after this set?

• If two or more answers showed the same pattern in the table, repeat only those questions tomorrow with new numbers. Do not sit a full test yet.
• For mixed OC practice with worked solutions, open the OC practice tests guide.
• For full papers to sit under time, see the OC past papers page.
• The free mock tests need a free classroom account. Create one only when your child is ready for a timed sitting.
• If you want a structured set of full OC tests after that, see what the OC course includes and check it matches your child's entry year before buying.

Sources and review

Official source (checked 2026-10-15): NSW Department of Education, Opportunity Class practice tests: the test is computer-based with three multiple-choice sections. The Department does not list counting as a named topic.

Where the questions came from: all eight problems and both models are original Braintree Coaching Australia material written for this page. None is taken from an official or third-party paper. The problems use four options for simplicity.

Author: James Rush, content writer, Braintree Coaching Australia.

Answer check: every answer was solved by listing every case, without the multiplication shortcut. A named teacher review has not yet been recorded for this set. If you find an error, please tell us.

Last updated: 2026-10-15.