⚡ Quick Memory Keys
Pin these before you start — each card is a core formula you'll need.
PERMUTATION
"ORDER matters"
\(P(n,r) = \dfrac{n!}{(n-r)!}\)
Pick r from n, sequence counts
Pick r from n, sequence counts
COMBINATION
"ORDER irrelevant"
\(C(n,r) = \dbinom{n}{r} = \dfrac{n!}{r!(n-r)!}\)
Pick r from n, group only
Pick r from n, group only
PROBABILITY
"Favorable / Total"
\(P(A) = \dfrac{|A|}{|S|}\)
\(0 \le P(A) \le 1\)
\(0 \le P(A) \le 1\)
COMPLEMENT
"NOT = 1 minus"
\(P(A') = 1 - P(A)\)
Easier to count what's excluded
Easier to count what's excluded
ADDITION RULE
"OR = add, minus overlap"
\(P(A \cup B) = P(A)+P(B)-P(A \cap B)\)
MULTIPLICATION
"AND = multiply"
Independent: \(P(A \cap B) = P(A) \cdot P(B)\)
📦 Topic 1 — Set Theory
★ Example
Q 01
Let \(U = \{1,2,3,4,5,6,7,8\}\), \(A = \{2,4,6,8\}\), \(B = \{3,4,5,6\}\).
Find \(|A \cup B|\) — the number of elements in A or B.
Find \(|A \cup B|\) — the number of elements in A or B.
UNION = everything in A or B (no duplicates) · |A∪B| = |A| + |B| − |A∩B|
Q 02
In a class of 30 students, 18 play football, 14 play basketball, and 5 play both. How many students play neither sport?
NEITHER = Total − |F∪B| · and |F∪B| = |F|+|B|−|F∩B|
Q 03
\(A = \{x \in \mathbb{Z} : 1 \le x \le 5\}\), \(B = \{x \in \mathbb{Z} : 3 \le x \le 7\}\).
Which set equals \(A \cap B\)?
Which set equals \(A \cap B\)?
INTERSECTION ∩ = elements in BOTH sets simultaneously
🔢 Topic 2 — Permutations
★ Example
Q 04
In how many ways can 5 students line up for a photograph?
\[P(n, n) = n! \quad\Rightarrow\quad 5! = 5 \times 4 \times 3 \times 2 \times 1\]
FACTORIAL: n! = n×(n−1)×…×1 · "Shrinking choices at each position"
Q 05
How many 3-digit numbers can be formed using digits \(\{1, 2, 3, 4, 5\}\) with no repetition?
P(n,r) = n!/(n−r)! · Choosing 3 positions from 5 digits, ORDER matters
Q 06
The word LEVEL has 5 letters. How many distinct arrangements exist?
⚠️ Watch out: repeated letters reduce the count!
Formula: \(\dfrac{n!}{\text{(repeated letter counts)}!}\)
Formula: \(\dfrac{n!}{\text{(repeated letter counts)}!}\)
REPEATED LETTERS: divide by factorial of each repeated letter count
🎯 Topic 3 — Combinations
★ Example
Q 07
A team of 3 players must be chosen from a group of 7. How many ways can this be done?
\[\binom{7}{3} = \frac{7!}{3! \cdot 4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35\]
COMBINATION: "choose" = C(n,r) — order does NOT matter · team ≠ lineup
Q 08
Which expression equals \(\dbinom{8}{5}\)?
SYMMETRY RULE: C(n,r) = C(n, n−r) · "choosing 5 from 8 = leaving out 3"
Q 09
A student must choose 4 questions from 6 in Section A and 3 questions from 5 in Section B. In how many ways can this be done?
MULTIPLY independent choices: C(6,4) × C(5,3)
Q 10
From 4 boys and 3 girls, a committee of 4 is formed. How many ways include exactly 2 girls?
SPLIT selection: C(3,2) for girls × C(4,2) for boys
🎲 Topic 4 — Basic Probability
★ Example
Q 11
A card is drawn at random from a standard deck of 52 cards. What is the probability of drawing a King?
\[P(\text{King}) = \frac{\text{number of Kings}}{\text{total cards}} = \frac{4}{52} = \frac{1}{13}\]
P = favorable / total · Always simplify your fraction!
Q 12
A bag has 3 red, 5 blue, and 2 green balls. What is the probability of not picking a blue ball?
COMPLEMENT: P(not blue) = 1 − P(blue) · faster than counting non-blue
Q 13
A fair die is rolled. What is \(P(\text{even} \cup \text{greater than 4})\)?
ADDITION RULE: P(A∪B) = P(A)+P(B)−P(A∩B) · don't double-count overlap!
Q 14
Two fair coins are tossed. What is the probability of getting at least one Head?
AT-LEAST-ONE trick: P(≥1 Head) = 1 − P(0 Heads) = 1 − P(TT)
🔗 Topic 5 — Conditional & Combined
★ Example
Q 15
Box A has 3 red and 2 white balls. Two balls are drawn without replacement.
What is \(P(\text{both red})\)?
\[P(\text{both red}) = \frac{3}{5} \times \frac{2}{4} = \frac{6}{20} = \frac{3}{10}\]
WITHOUT REPLACEMENT: denominator shrinks on 2nd draw · multiply conditional probs
Q 16
\(P(A) = 0.4\), \(P(B) = 0.5\), \(P(A \cap B) = 0.2\).
Find \(P(A | B)\) — the probability of A given B has occurred.
Find \(P(A | B)\) — the probability of A given B has occurred.
CONDITIONAL: P(A|B) = P(A∩B) / P(B) · "restrict sample space to B"
Q 17
Events A and B are independent. \(P(A) = 0.3\), \(P(B) = 0.6\).
What is \(P(A \cup B)\)?
What is \(P(A \cup B)\)?
INDEPENDENT: P(A∩B) = P(A)·P(B) first, then use addition rule for union
Q 18
A hand of 5 cards is dealt from a standard 52-card deck. How many different hands are possible?
CARD HAND = combination (order irrelevant) · C(52, 5)
Q 19
A password consists of 4 distinct digits chosen from \(\{0,1,2,...,9\}\). How many passwords are possible if the first digit cannot be 0?
RESTRICTION FIRST: fix the restricted position first, then fill remaining
Q 20
A group of 10 people includes 4 doctors. A committee of 3 is randomly selected.
What is the probability the committee contains exactly 1 doctor?
Hint: P = (ways to choose exactly 1 doctor × ways to choose other 2) ÷ (total ways to choose 3)
HYPERGEOMETRIC setup: C(4,1)×C(6,2) / C(10,3)
🎉
Quiz Complete!
0/20
Keep studying and try again.