20 carefully designed questions — from the basics of counting principles to tricky permutation problems. Read each memory tip before answering!
⚡ Quick Reference
Factorial
n! = n×(n−1)×…×1
KEY WORD: "arrange all"
Permutation
ₙPᵣ = n!/(n−r)!
KEY WORD: "order matters"
Combination
ₙCᵣ = n!/r!(n−r)!
KEY WORD: "order doesn't matter"
Counting Prin.
a × b × c …
KEY WORD: "AND → multiply"
Counting Principle
Questions 1–5
Q 01
Counting Principle
A Dunkin' Donuts has 3 sizes, 2 dairy options, and 2 sweetener options for Cold Brew.
How many different Cold Brew orders are possible?
⚡ Memory: AND → × · Each choice multiplies the total
Q 02
Counting Principle
Coffee at Dunkin' has 4 sizes, 7 dairy options, 8 flavor options, and 2 sweetener options.
How many different coffee orders exist?
⚡ Memory: MULTIPLY ALL BRANCHES · Tree diagram = one branch per category
Q 03
Counting Principle
A school café offers 5 sandwiches, 4 drinks, and 3 desserts.
How many different lunch combos (one of each) can a student make?
⚡ Memory: COMBO = × · "One of each" is always multiply
Q 04
Counting Principle
A license plate has 3 letters followed by 3 digits. Letters and digits can repeat.
How many different license plates are possible?
(26 letters, 10 digits)
⚡ Memory: REPEAT ALLOWED → same number each slot
Q 05
Counting Principle
A pizza shop offers 2 crust types, 3 sauce choices, and 6 toppings.
A customer picks exactly 1 crust, 1 sauce, and 1 topping.
How many different pizzas?
⚡ Memory: TREE = multiply down each level
Factorial
Questions 6–9
Q 06
Factorial
What is the value of 5!?
⚡ Memory: n! = n × (n−1) × … × 1 · Just multiply down to 1
Q 07
Factorial
In how many ways can 6 students line up in a single row?
⚡ Memory: ARRANGE ALL n items = n! · First slot has most choices
Q 08
Factorial
Simplify:
7! ÷ 5!
⚡ Memory: CANCEL DOWN · 7!/5! = 7×6 only!
Q 09
Factorial
Which of the following is NOT equal to 24?
⚡ Memory: 4! = 24 · Know your small factorials cold: 0!=1, 1!=1, 2!=2, 3!=6, 4!=24
Permutations
Questions 10–15
Q 10
Permutation
Using the digits 1–6 with no repeats, how many different 6-digit passcodes can be created?
(This is the "Win free coffee for life!" problem from class.)
⚡ Memory: ALL digits, no repeat = 6!
Q 11
Permutation
Using digits 1–6 with no repeats, how many different 4-digit passcodes can be made?
(The "Win free coffee for a year!" problem.)
⚡ Memory: ₆P₄ = 6×5×4×3 · Start from n, count down r times
Q 12
Permutation
What is the value of ₈P₃?
₈P₃ = 8! ÷ (8−3)! = ?
⚡ Memory: ₙPᵣ = n×(n−1)×… · Multiply r terms starting from n
Q 13
Permutation
There are 10 runners in a race. How many ways can 1st, 2nd, and 3rd place be awarded?
(Hint: Does the order of the medals matter?)
⚡ Memory: RANK/PLACE = ORDER MATTERS = Permutation
Q 14
Permutation
🚨 Tricky! Which expression is the correct formula for ₙPᵣ?
⚡ Memory: P for PLACE · denominator = (n−r)! · Don't mix with Combination!
Q 15
Permutation
A club needs to elect a President, Vice-President, and Secretary from 9 members.
No one can hold two positions. How many outcomes are possible?
⚡ Memory: DIFFERENT ROLES = ORDER MATTERS = ₙPᵣ
Combinations & Mixed
Questions 16–20
Q 16
Combination
🚨 Classic Trap! A teacher needs to choose 3 students from a class of 10 to work on a group project.
How many groups are possible?
(Nobody has a special role — they're all just "in the group.")
⚡ Memory: GROUPS/SELECT/CHOOSE → Combination · order doesn't matter
Q 17
Combination
🚨 P vs C Decision! You have 8 books. Which scenario uses a Combination (not Permutation)?
⚡ Memory: P = Placement (1st/2nd/3rd) · C = Collection (grab a bunch)
Q 18
Combination
Calculate ₇C₂.
₇C₂ = 7! / [2! × (7−2)!] = ?
⚡ Memory: C has an EXTRA r! in the denominator · That's what "removes order"
Q 19
Probability
🚨 Hardest! Digits 1–6 are used to create a random 4-digit passcode (no repeats).
What is the probability that your passcode matches exactly one specific target?
⚡ Memory: P(event) = favorable / total · Total = ₆P₄ = 360
Q 20
Mixed Review
🚨 Final Boss! Ian needs to explain to a friend: "Why is ₅P₃ bigger than ₅C₃?"
Which explanation is correct?
⚡ Memory: ₙPᵣ = ₙCᵣ × r! · P is always ≥ C · Permutation counts every ordering