Permutation, Combination & Probability

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Counting Principles: Product & Sum Rules

⏱ 4 min read🧩 6 question types🎯 12 practice Q

The idea in one minute

Almost every counting question is built from two rules. When steps happen one after another ('and'), multiply the choices. When separate cases work ('or'), add them. Numbers and codes are built slot by slot.

What counting means

A counting question asks "in how many ways". A way is one complete choice or one complete outcome.

A menu has 3 drinks and 4 snacks. Picking one drink and one snack is one way. There are 12 such ways.

Rule: Every counting question is built from just two rules — multiply for "and", add for "or".

The product rule

Two steps happen one after another. Step 1 has $m$ options. Step 2 has $n$ options. The pair happens in $m \times n$ ways.

A canteen has 2 shirts and 3 trousers. Outfits $= 2 \times 3 = 6$ .

Three steps work the same way. 2 shirts, 3 trousers and 2 caps give $2 \times 3 \times 2 = 12$ outfits.

Tip: Underline the joining word in the question. "Shirt and trousers" means multiply.

The sum rule

Two separate cases can happen, but only one will. Case 1 has $m$ ways. Case 2 has $n$ ways. The total is $m + n$ .

A town has 4 direct flights and 3 direct trains to another town. Travel options $= 4 + 3 = 7$ .

Watch: Add only when the cases cannot happen together. Overlapping cases count some outcomes twice.

Numbers with repetition allowed

A code or a number is built slot by slot. Each slot is one step.

Digits 1, 2, 3, 4 make 3-digit codes with repetition allowed. Each slot has 4 options: $4 \times 4 \times 4 = 64$ codes.

With $n$ distinct items and $r$ slots, repetition allowed gives $n^r$ outcomes. A 4-digit PIN from ten digits: $10^4 = 10000$ .

Numbers without repetition

No digit may be used twice. The first slot has $n$ options. The next has $n - 1$ . Then $n - 2$ , and so on.

3-digit numbers from 1, 2, 3, 4, 5 without repetition: $5 \times 4 \times 3 = 60$ .

Each such arrangement is a permutation. The permutation subtopic continues exactly from here.

When zero is in the digits

Zero cannot lead a number. Fill the restricted slot first.

Digits 0, 1, 2, 3 give 3-digit numbers without repetition. First slot: 3 options (1, 2, 3). Second slot: 3 (zero plus the two unused digits). Third: 2. Total $3 \times 3 \times 2 = 18$ .

Watch: $4 \times 3 \times 2 = 24$ counts strings like 012, which are not 3-digit numbers.

Mixing the two rules

Long questions chain both rules. Split the situation into cases first. Add the cases. Use the product rule inside each case.

From A to C: 4 direct flights, or a bus then a train through B (3 buses, 2 trains). Total $= 4 + 3 \times 2 = 10$ ways.

Example: Numbers of 1 to 3 digits from 1, 2, 3 without repetition: $3 + 3 \times 2 + 3 \times 2 \times 1 = 15$ .

Count each case fully before adding. Half-built cases are the main source of errors here.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Multi-step choices (product rule)

How to spot it:

Two or more choices happen one after another — shirts and trousers, a route from A to B and then B to C.

m \times n

Method

Spot the independent steps.
Count the options for each step.
Multiply all the counts together.

Why it works:

Each option of step 1 pairs with each option of step 2, so the counts multiply.

Try this

A man has 2 shirts, 3 trousers and 2 caps. How many different shirt-trouser-cap outfits can he wear?

Show solution

Outfits $= 2 \times 3 \times 2$ .
$= 12$ .

Answer

Type 2very common2 practice Q

Numbers or codes with repetition allowed

How to spot it:

'Digits/letters may be repeated', or the question says nothing about repeating.

n^r

Method

Count the available digits or letters ( $n$ ).
Count the slots ( $r$ ).
Each slot has $n$ choices, so the answer is $n^r$ .
For a number, keep zero out of the first slot.

Why it works:

Every slot is filled independently, so the counts multiply into a power.

Try this

How many 3-digit numbers can be formed using 1, 2, 3, 4 if digits may repeat?

Show solution

Each slot has 4 choices.
$4^3 = 64$ .

Answer

Type 3very common2 practice Q

Numbers without repeating a digit

How to spot it:

'Without repeating any digit' appears in the question.

n(n-1)(n-2)\ldots(n-r+1)

Method

Give the first slot $n$ options.
Give each later slot one option fewer.
Multiply the slot counts.
For $r$ slots the last factor is $n - r + 1$ .

Why it works:

A used digit cannot return, so every later slot loses one option.

Try this

How many 3-digit numbers can be formed from 1, 2, 3, 4, 5 without repetition?

Show solution

Slots: $5 \times 4 \times 3$ .
$= 60$ .

Answer

Type 4common2 practice Q

Numbers with a property (even, odd, greater than)

How to spot it:

'How many are even / odd / greater than…' — a condition on the first or last digit.

Method

Fill the most restricted slot first.
Then fill the remaining slots.
Multiply the slot counts.
Check the condition is truly enforced.

Why it works:

The condition cuts the options of one slot, and the rest of the slots follow.

Try this

How many 3-digit even numbers can be formed from 1, 2, 3, 4 without repetition?

Show solution

Units: 2 or 4 → 2 options.
Hundreds: 3 left; tens: 2 left.
$2 \times 3 \times 2 = 12$ .

Answer

Type 5common2 practice Q

Sum rule over separate cases

How to spot it:

'Or' joins two routes or cases, or the journey/items come in disjoint types.

m + n

Method

Split the outcomes into cases that cannot overlap.
Count each case with the product rule.
Add the case counts.

Why it works:

Cases that cannot happen together never share an outcome, so their counts simply add.

Try this

To reach town C from town A: 4 direct flights, or a bus from A to B and then a train from B to C (3 buses, 2 trains). How many travel options are there?

Show solution

Direct: 4.
Via B: $3 \times 2 = 6$ .
$4 + 6 = 10$ .

Answer

Type 6common2 practice Q

Zero cannot be the first digit

How to spot it:

0 is among the given digits and numbers (not codes) are asked.

Method

Fill the first slot first, skipping zero.
Fill the other slots from what remains.
Multiply the slot counts.

Why it works:

A leading zero makes a shorter number, so zero is banned from the first slot.

Try this

How many 3-digit numbers can be formed from 0, 2, 3, 4 without repetition?

Show solution

First slot: 3 options (2, 3, 4).
Second: 3; third: 2.
$3 \times 3 \times 2 = 18$ .

Answer

Formula sheet

Product rule

m \times n

$m$ options for step 1 and $n$ for step 2, joined by 'and'.

Sum rule

m + n

Two cases that cannot happen together, joined by 'or'.

Strings with repetition

n^r

$r$ slots from $n$ items, repeats allowed.

Arrangements without repetition

n(n-1)(n-2)\ldots(n-r+1)

$r$ ordered slots, no item reused.

Shortcuts that save time

⚡ Fill the restricted slot first

Even numbers, numbers ending in 5, no leading zero — the condition always lives in one slot. Fill that slot first, then the rest.

Example

How many 3-digit numbers from 1, 2, 3, 4, 5 (no repetition) end in 5?

Show solution

Units: only 5 — 1 way.
Hundreds: 4; tens: 3.
$4 \times 3 = 12$ .

Answer

⚡ AND multiply, OR add — say it out loud

Read the question and underline the joining words. 'And' multiplies counts; 'or' adds cases. Most counting errors are one wrong joining word.

Example

A to B: 2 roads. B to C: 3 roads. A to C direct: 4 flights. How many ways from A to C?

Show solution

Via B: $2 \times 3 = 6$ .
Direct: 4.
$6 + 4 = 10$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Adding steps instead of multiplying — 3 choices and then 4 choices make $12$ outcomes, not $7$ .

Mistake 02

Letting zero lead the number — fill the first slot before the others whenever 0 is among the digits.

Mistake 03

Reading 'without repetition' as with repetition — $5^3 = 125$ and $5 \times 4 \times 3 = 60$ answer two different questions.

Mistake 04

Adding overlapping cases — 'flights or morning departures' share some flights, so the plain sum double-counts.

Mistake 05

Forgetting the condition in the middle of the work — re-check the restricted slot before answering.

Quick revision

Read this the night before the exam.

'And' multiplies; 'or' adds.
Repetition allowed: $n^r$ outcomes for $r$ slots.
No repetition: $n(n-1)(n-2)\ldots$
Zero never leads a number — fill the first slot first.
Fill the most restricted slot first in every question.
Split into non-overlapping cases, then add the cases.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 8 min · wrong answers go to your mistake notebook automatically.

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