Core Concepts & Formulas
Study these concepts before attempting the quiz.
1Fundamental Counting Principle (FCP)
If one event can occur in m ways and a second independent event can occur in n ways, then the two events together can occur in m × n ways.
Total ways = m₁ × m₂ × m₃ × ⋯ × mₖ
⭐ Remember
Multiply when choices are
independent. Add when choices are
mutually exclusive alternatives.
Example
A restaurant has 4 appetizers, 5 main courses, and 3 desserts. How many different 3-course meals are possible?
4 × 5 × 3 = 60 meals
2Permutations (Order Matters)
A permutation is an arrangement of objects where order matters.
P(n, r) = n! / (n − r)!
P(n, n) = n! (all n objects)
P(n, 1) = n (choose 1)
⭐ Key Formulas
n! = n×(n−1)×⋯×2×1
0! = 1
1! = 1
Example
How many ways can 5 students be arranged in a row of 3 seats?
P(5,3) = 5!/(5−3)! = 5×4×3 = 60 ways
3Combinations (Order Doesn't Matter)
A combination is a selection of objects where order does not matter.
C(n, r) = n! / [r! × (n − r)!]
C(n, r) = C(n, n−r) (symmetry)
C(n, 0) = C(n, n) = 1
⭐ Permutation vs Combination
P(n,r) = C(n,r) × r!
Combinations = Permutations ÷ r! (remove the ordering)
Example
From 6 people, how many ways to choose a team of 4?
C(6,4) = 6!/(4!×2!) = 15 ways
4Basic Probability
Probability measures the likelihood of an event occurring.
P(A) = (number of favorable outcomes) / (total equally likely outcomes)
0 ≤ P(A) ≤ 1
P(A) + P(A') = 1 [complement rule]
⭐ Remember
P(A') = 1 − P(A). Often easier to calculate complement!
Example
A bag has 3 red and 5 blue balls. What is P(red)?
P(red) = 3/8
5Addition Rule (Union of Events)
For the probability that event A or event B occurs:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
If A and B are mutually exclusive: P(A ∩ B) = 0
→ P(A ∪ B) = P(A) + P(B)
Example
P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2. Find P(A ∪ B).
P(A ∪ B) = 0.4 + 0.5 − 0.2 = 0.7
6Conditional Probability & Independence
P(A | B) = P(A ∩ B) / P(B) [conditional]
Independent events: P(A ∩ B) = P(A) × P(B)
P(A | B) = P(A) when A and B are independent
⭐ Independence Test
Check: P(A ∩ B) = P(A) · P(B). If equal → independent.
Example
P(A∩B) = 1/6, P(B) = 1/3. Find P(A|B).
P(A|B) = (1/6) ÷ (1/3) = 1/2
7Circular Permutations & Repetition
Circular arrangements of n objects = (n − 1)!
Permutations with repetition (n objects, r chosen) = nʳ
Permutations of n with repeated elements:
n! / (n₁! × n₂! × ⋯ × nₖ!)
⭐ Circular vs Linear
Circular: fix one object, arrange the rest → (n−1)!
Necklace (flip = same): (n−1)! / 2
Example
In how many ways can 5 people sit around a round table?
(5−1)! = 4! = 24 ways