Counting Principles: Addition Rule, Multiplication Rule, and Combinatorics

2026.04.24

by QANDA

Counting principles form the foundation of probability and combinatorics. Starting from middle school and extending through advanced statistics, these concepts help you calculate the number of possible outcomes in any scenario.

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Counting is the starting point for all probability problems. Master the difference between the addition rule and multiplication rule, and between permutations and combinations, and you'll solve most problems with ease.

Fundamental Counting Principles

What Is Counting?

Counting in mathematics means determining the total number of possible outcomes of an event. For example, rolling a single die has 6 possible outcomes.

Addition Rule vs Multiplication Rule

Property	Addition Rule	Multiplication Rule
When to use	Events are mutually exclusive	Events happen in sequence
Keywords	"or", "either... or"	"and", "then", "consecutively"
Operation	$m + n$	$m \times n$

Permutations and Combinations

Step 1: Permutations

A permutation counts the ways to select $r$ items from $n$ and arrange them in order.

$_nP_r = \frac{n!}{(n-r)!}$

Example: Choosing a president and vice president from 5 people:

$_5P_2 = 5 \times 4 = 20$

Step 2: Combinations

A combination counts the ways to select $r$ items from $n$ with no regard to order.

$_nC_r = \frac{n!}{r!(n-r)!} = \frac{_nP_r}{r!}$

Example: Choosing a committee of 2 from 5 people:

$_5C_2 = \frac{5 \times 4}{2 \times 1} = 10$

Step 3: How to Tell Them Apart

The key question: Does the order matter?

"Arrange", "assign roles", "rank" → Permutation
"Select", "choose a team", "pick" → Combination

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Confusing permutations and combinations is the most common error. Ask yourself: "If I swap two items, is it a different outcome?" If yes, use permutations. If no, use combinations.

Advanced: Special Cases

Permutations with Repetition

When $n$ items include $p$ identical items of one type, $q$ of another, and $r$ of another:

$\frac{n!}{p! \cdot q! \cdot r!}$

Example: Arrangements of the letters in "APPLE" (2 P's):

$\frac{5!}{2!} = 60$

Circular Permutations

Arranging $n$ items in a circle:

$(n-1)!$

Practice Problems

Example 1: Addition and Multiplication Rules

Problem: There are 3 trains and 5 KTX services from Seoul to Busan. Find (a) the number of one-way options and (b) the number of round-trip options.

Show Solution

(a) One-way (addition rule):

Choose train OR KTX:

$3 + 5 = 8 \text{ options}$

(b) Round trip (multiplication rule):

Going and returning are sequential events:

$8 \times 8 = 64 \text{ options}$

Answer: (a) 8, (b) 64

Example 2: Permutation vs Combination

Problem: From 7 students, find (1) the number of ways to line up 3 students, and (2) the number of ways to form a team of 3.

Show Solution

(1) Line up (order matters → permutation):

$_7P_3 = 7 \times 6 \times 5 = 210$

(2) Form a team (order doesn't matter → combination):

$_7C_3 = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$

Answer: (1) 210, (2) 35

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When in doubt, start with a small example. Try choosing 2 from 3 items and list all possibilities by hand — it immediately clarifies whether order matters.

Top 3 Common Mistakes

Mixing up permutations and combinations — "Selection" is combination, but "arrangement" is permutation. Look for whether order changes the outcome.
Confusing addition and multiplication rules — "Or" means add, "and" means multiply. Identify the keywords in the problem.
Forgetting $0! = 1$ — Remember that $_nC_0 = 1$ and $_nC_n = 1$ .

Related Concepts

Probability — counting → probability
Sequences and Series — patterns
Quadratic Equations — algebra skills

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