The Multiplication and Addition Principles
Key Terms
- Multiplication principle
- If task 1 has m outcomes AND task 2 has n outcomes (independently), the combined count is m × n.
- Addition principle
- If event A can occur in m ways OR event B in n ways, and they are mutually exclusive, the total is m + n.
- Factorial
- n! = n × (n−1) × … × 2 × 1; counts all ordered arrangements of n distinct objects. 0! = 1.
- Mutually exclusive
- Two events that cannot both occur; when these are the alternatives, ADD the counts.
- Tree diagram
- A visual tool listing all outcomes of a sequential counting process.
- Complementary counting
- Total possible outcomes − unwanted outcomes = wanted outcomes.
Core Principles
Multiplication Principle (AND): If task 1 can be done in m ways and task 2 can be done in n ways independently, then both tasks together can be done in m × n ways.
Addition Principle (OR): If event A can occur in m ways and event B can occur in n ways, and A and B are mutually exclusive (cannot both happen), then A or B occurs in m + n ways.
Factorial: n! = n × (n−1) × (n−2) × … × 2 × 1 and 0! = 1
| Keyword | Operation | Example |
|---|---|---|
| AND — sequential choices, both happen | Multiply × | Choose a shirt AND pants: 4 × 3 = 12 |
| OR — mutually exclusive alternatives | Add + | Travel by bus OR train: 3 + 5 = 8 |
Worked Example 1 — Menu Combinations
A restaurant offers 3 entrees, 4 mains and 2 desserts. How many different 3-course meals are possible?
Each course is chosen independently (AND reasoning):
Total = 3 × 4 × 2 = 24 meals
Worked Example 2 — Number Plates
A number plate has 3 letters followed by 3 digits (0–9). No letter repeats; no digit repeats within its section. How many plates are possible?
Letters: 26 × 25 × 24 = 15 600 Digits: 10 × 9 × 8 = 720
Total = 15 600 × 720 = 11 232 000 plates
Why Does AND Mean Multiply?
When you make sequential, independent choices, the number of outcomes multiplies. Imagine choosing an outfit: if you have 4 shirts and 3 pants, for each of the 4 shirts you can pair any of the 3 pants — giving 4 rows of 3 options each, which is 4 × 3 = 12 outfits. You are counting all cells in a grid.
The key condition is independence: the number of options for task 2 must not depend on which option was chosen for task 1.
Why Does OR Mean Add?
When two events are mutually exclusive — they cannot occur simultaneously — you simply count each group and add them. If there are 3 red balls and 5 blue balls in a bag, there are 3 + 5 = 8 ways to pick one ball. Picking red and picking blue cannot both happen at once, so we add.
Warning: If events can overlap (e.g., "pick a card that is red OR a king"), simple addition overcounts. You must then use the Inclusion-Exclusion Principle (covered in a later lesson).
Factorial as Multiplication Principle
Suppose you have n distinct objects to arrange in a row. There are n choices for position 1, then n−1 remaining for position 2, then n−2 for position 3, and so on. By the multiplication principle:
n × (n−1) × (n−2) × … × 1 = n!
So factorial is simply the multiplication principle applied to filling positions one at a time, reducing options by one each time.
Key Values to Know
0! = 1 (by convention — there is exactly one way to arrange zero objects: do nothing)
1! = 1 2! = 2 3! = 6 4! = 24 5! = 120 6! = 720 7! = 5040
Simplifying Factorial Expressions
You rarely need to compute large factorials in full. Instead, cancel common factors:
8! / 6! = (8 × 7 × 6!) / 6! = 8 × 7 = 56
Always write out just enough terms until the fraction cancels cleanly.
Mastery Practice
-
Fluency
Q1 — 3-Course Meals
A restaurant offers 4 starters, 6 mains and 3 desserts. How many different 3-course meals are possible?
-
Fluency
Q2 — Factorial Calculations
Calculate: (a) 5! (b) 0! (c) 8!/6! (d) 10!/(2! × 8!)
-
Fluency
Q3 — PIN Codes
How many 4-digit PIN codes are possible if: (a) digits can repeat? (b) digits cannot repeat?
-
Fluency
Q4 — Coin Sequences
A coin is tossed 4 times. How many possible sequences of heads and tails are there?
-
Understanding
Q5 — Number Plates
A standard number plate has 3 letters followed by 3 digits (0–9). Letters may not repeat within the letter section, and digits may not repeat within the digit section. How many number plates are possible?
-
Understanding
Q6 — Committee Roles
In how many ways can a president, vice-president and secretary be chosen from a group of 10 people? (The same person cannot hold two roles.)
-
Understanding
Q7 — Subject Selection
A student must choose one subject from each of three groups: Group A (3 options), Group B (4 options), Group C (2 options). How many different subject combinations are possible?
-
Understanding
Q8 — Pizza Options
A pizza shop offers 3 sizes, 2 types of base, 5 meats and 8 vegetable toppings. How many different pizzas can be made if: (a) no toppings are selected (b) exactly one meat is chosen (c) exactly one topping of any kind is chosen?
-
Problem Solving
Q9 — 3-Digit Numbers
How many 3-digit numbers (100–999) are: (a) even? (b) divisible by 5? (c) have all different digits?
-
Problem Solving
Q10 — Batting Order with Restrictions
A cricket team of 11 players must be arranged in a batting order. The captain must bat first and the vice-captain must bat last. How many batting orders are possible?