Questions: Counting Principles and Multiplication Principle
5 questions to test your understanding
Score: 0 / 5
Question 1 Multiple Choice
A restaurant offers a lunch special: choose 1 soup from 4 options OR 1 sandwich from 6 options (you pick exactly one dish). A set meal requires choosing 1 soup AND 1 sandwich. How many distinct selections are available under each option?
CBoth have 10 options — you are making one selection from two categories in each case
DBoth have 24 options — the categories are independent so you always multiply
The lunch special presents mutually exclusive alternatives — you pick one thing from a combined pool of 4 + 6 = 10 dishes (addition principle). The set meal requires sequential independent choices — one soup AND one sandwich — giving 4 × 6 = 24 combinations (multiplication principle). The central skill in combinatorics is recognizing which principle applies: add when choosing between alternatives (one or the other), multiply when making sequential independent selections (one of each). Confusing the two is the most common error.
Question 2 Multiple Choice
A password must be exactly 3 characters, with each character independently chosen from the digits 0–9 (repetition allowed). How many possible passwords exist?
A30 — adding 10 options for each of the 3 positions
B1,000 — multiplying 10 × 10 × 10, one independent choice per position
C720 — using 10 × 9 × 8 because repeated digits must be avoided
D10 — since there are only 10 distinct digits available
Each of the 3 positions is an independent sequential choice with 10 options (0–9), and repetition is allowed, so each position always has 10 options regardless of previous choices. The multiplication principle gives 10 × 10 × 10 = 1,000. Option A (adding 10 three times = 30) confuses sequential choices with mutually exclusive alternatives — addition would be correct if you were choosing the password length (either 1 digit, or 2 digits, or 3 digits). Option C (10 × 9 × 8 = 720) would apply if repetition were forbidden — a different problem.
Question 3 True / False
If a task has two stages with m choices at stage one and n choices at stage two, the total number of outcomes is generally m × n.
TTrue
FFalse
Answer: False
False — multiplication applies only when the stages are sequential and independent. If the tasks are mutually exclusive alternatives (you do one or the other, not both), you add: m + n total options. Additionally, if choices at stage two depend on what was chosen at stage one (e.g., 'pick a letter then pick a different letter'), n may not be constant — and you still multiply, but with the actual number of choices at each step. The multiplication principle requires sequential structure; the independence condition determines whether n is constant or varies.
Question 4 True / False
The addition principle applies when you are choosing one item from two disjoint sets of options (either/or), while the multiplication principle applies when you are making one selection from each of multiple independent sets (one of each).
TTrue
FFalse
Answer: True
True. This is the core distinction. Addition is for disjoint alternatives: 'I will do task A OR task B.' The total outcomes are the number of ways to do A plus the number of ways to do B — provided the sets are mutually exclusive (no overlap). Multiplication is for sequential selections: 'I will do task A AND THEN task B.' The total outcomes are the product of the choices at each step. The diagnostic question to ask in any counting problem: are you making a sequence of choices (each happening independently), or choosing among alternatives (picking exactly one option from a menu)?
Question 5 Short Answer
Explain the key difference between when to add and when to multiply in counting problems, using one concrete example of each.
Think about your answer, then reveal below.
Model answer: Add when the choices are mutually exclusive alternatives — you select exactly one option from a combined pool. Example: a store sells 4 red shirts and 6 blue shirts; if you buy exactly one shirt, you have 4 + 6 = 10 choices. Multiply when you make a sequence of independent selections — one choice from each category. Example: you buy one shirt (4 options) AND one pair of pants (3 options); the number of outfits is 4 × 3 = 12. The diagnostic test: ask 'am I choosing between options (or/or) or combining options from different categories (and/and)?' Or/or → add. And/and → multiply.
Many real problems disguise which principle applies. The phrase 'how many ways can you...' often hides whether you are doing both things or choosing between them. Drawing a slot for each independent decision — and labeling it with how many options it has — then multiplying all the slots is a reliable technique for sequential problems. Switching to addition when the slots represent alternatives (not sequences) is the correction.