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You will learn: Choose between addition and multiplication, and distinguish ordered choices from unordered pairs.

Before you start: Multiplication and small systematic lists.

Describe exactly what one outcome is

Before counting, decide whether order matters, whether repetition is allowed, and whether there are special restrictions. Choosing a captain and a deputy is different from choosing an unnamed pair: swapping the two students changes the first outcome but not the second.

Two useful rules

Multiply for consecutive choices. If each of a first choices can be followed by b second choices, there are ab ordered outcomes. For example, 3 shirts and 2 hats give 3 × 2 = 6 outfits, provided every shirt can go with every hat.

Add for separate cases. If an outcome belongs to exactly one of two cases, and the cases contain a and b outcomes, their total is a + b. The “exactly one” condition matters: overlapping cases would count some outcomes twice.

Three shirts and two hats
ShirtHat XHat Y
AAXAY
BBXBY
CCXCY
Every outfit appears in one cell, so none is missed or counted twice.

Worked example 1: a restriction changes the choices

How many three-digit numbers can be formed from 0, 1, 2, 3 without repeating a digit? The first digit cannot be 0, so it has 3 choices. After it is chosen, 3 digits remain for the second position, including 0. Then 2 remain for the last position. Thus there are 3 × 3 × 2 = 18 numbers. Using 4 choices for the first digit would incorrectly include strings such as 012, which are not three-digit numbers.

Worked example 2: an unordered pair

How many pairs can be selected from five students? First choose one student and then a different student: there are 5 × 4 = 20 ordered choices. But each pair appears twice, once in each order. Divide by 2 to get 10 pairs. We may divide because every pair has been counted exactly twice.

For n distinct objects, the same argument gives n(n − 1)/2 unordered pairs. This is a derivation of the formula, not an extra rule to memorise without meaning.

Worked example 3: count what is missing

A three-character code uses A or B in each position. How many contain at least one A? There are 2³ = 8 codes altogether. Exactly one, BBB, contains no A. Therefore 8 − 1 = 7 contain at least one A. Counting the unwanted complement can be shorter than splitting the wanted codes into many cases.

A short checklist

State what your objects are. Explain the number of choices at each step. Check whether the choices depend on earlier decisions. If you divide to remove repeats, explain how many times each outcome was counted. A correct multiplication without these reasons may conceal a counting error.

Practise at your next step

A six-question session chooses from nine questions. Two correct answers in a row without hints move you up a level; an incorrect answer brings a simpler next question where one is available. Hints keep you at the same level. This is a practice suggestion, not an exam score or proof of mastery.

Interactive practice loads here. You can also use the complete question set below.

Write a proof of your own

Six students each shake hands once with every other student. Prove that there are 15 handshakes.

Write your reasoning on paper before comparing. The practice checker does not grade a written proof.

Compare your proof with a full solution

Each of the 6 students has 5 partners, producing 6 × 5 = 30 student-partner counts. Each handshake appears exactly twice: once from each participant’s point of view. Therefore the actual number of handshakes is 30/2 = 15. No student shakes hands with themselves, as required.

Check: did you state the assumptions, explain the key step, and reach the requested conclusion?

All nine practice questions, hints and solutions

This complete set works without the interactive practice. Hide each solution until you have made an attempt.

1. There are 4 shirts and 3 hats. Every shirt can be worn with every hat. How many shirt-and-hat outfits are possible?

Foundation

  1. 7
  2. 12
  3. 16
  4. 24
Hint

For each shirt there are three hat choices.

Answer and explanation

12. Each of the 4 shirts can be matched with any of the 3 hats. The outfit is determined by these two choices, so there are 4 × 3 = 12 outfits.

2. How many two-character codes use A or B in each position, with repetition allowed?

Foundation

  1. 2
  2. 3
  3. 4
  4. 6
Hint

List AA, then continue systematically.

Answer and explanation

4. The codes are AA, AB, BA and BB. There are 2 choices for each of 2 positions, giving 2 × 2 = 4.

3. Which task treats order as important?

Foundation

  1. Choosing an unnamed pair of students
  2. Choosing a captain and a deputy
  3. Choosing two different books as a set
  4. Choosing two distinct colours as a set
Hint

Would swapping the two choices create a different result?

Answer and explanation

Choosing a captain and a deputy. For captain and deputy, the roles differ: Ali as captain and Bo as deputy is different from Bo as captain and Ali as deputy. The unnamed sets in the other options do not change when their members are swapped.

4. How many two-digit numbers use digits 1, 2, 3, 4 without repetition?

Core

  1. 6
  2. 8
  3. 12
  4. 16
Hint

Choose the tens digit first, then a different units digit.

Answer and explanation

12. There are 4 choices for the tens digit and then 3 remaining choices for the units digit. Each number occurs once, so the total is 4 × 3 = 12.

5. How many unordered pairs can be chosen from 4 distinct students?

Core

  1. 4
  2. 6
  3. 8
  4. 12
Hint

Count ordered choices, then remove the double counting.

Answer and explanation

6. There are 4 × 3 = 12 ordered choices of two different students. Each unordered pair is counted in both orders, so there are 12/2 = 6 pairs.

6. A code has 3 characters, each A or B. How many codes contain at least one A?

Core

  1. 3
  2. 4
  3. 7
  4. 8
Hint

Subtract the codes with no A from all possible codes.

Answer and explanation

7. There are 2³ = 8 codes in total. Only BBB contains no A. Therefore 8 − 1 = 7 codes contain at least one A.

7. How many three-digit numbers use 0, 1, 2, 3 without repetition?

Stretch

  1. 12
  2. 18
  3. 24
  4. 27
Hint

Zero is forbidden only in the first position.

Answer and explanation

18. The first digit has 3 choices, excluding 0. The second has 3 remaining choices, now allowing 0. The last has 2. Thus the total is 3 × 3 × 2 = 18.

8. Seven students each shake hands once with every other student. How many handshakes occur?

Stretch

  1. 14
  2. 21
  3. 28
  4. 42
Hint

Every handshake has two participants.

Answer and explanation

21. There are 7 × 6 = 42 student-partner counts. Every handshake appears twice, once for each participant. Dividing by 2 gives 21 handshakes.

9. A shortest route on a grid from (0,0) to (2,2) uses only unit steps right or up. How many routes are possible?

Stretch

Two by two grid for the route questionStart at the lower-left point (0,0) and finish at the upper-right point (2,2). A sample shortest path takes right, up, right, up steps.(0,0)(2,2)
One sample route. Your task is to count every shortest route.
  1. 4
  2. 5
  3. 6
  4. 8
Hint

Every route has four steps, exactly two of which are right steps.

Answer and explanation

6. Choose the two positions of the right steps among four positions. There are 4 × 3 ordered choices, but each pair of positions appears twice, giving 6 routes. They are RRUU, RURU, RUUR, URRU, URUR and UURR.

Where to go next

Move on to the mixed proof-practice plan · Explore the wider Olympiad topic map

Teaching examples prepared for this guide, not attributed to a contest paper. Familiar elementary problems may appear in other learning materials. Report an unclear step or an error.