Probability & Combinations Guide: Permutations, Combinations & Probability Formulas
nCr = n! / (r!(n−r)!). Combinations count groups where order doesn't matter; permutations count arrangements where it does. Learn both with lottery and card examples.
The Fundamental Counting Principle
The multiplication principle: if event A can occur in m ways and event B can occur in n ways (independently), then both events together can occur in m × n ways. This principle underlies all counting formulas. Example: a restaurant has 4 appetizers, 6 entrees, and 3 desserts. Total three-course combinations = 4 × 6 × 3 = 72. For a 4-digit PIN with d
Permutations: When Order Matters
A permutation is an arrangement of items where the sequence matters. Choosing 1st, 2nd, and 3rd place is different from choosing a committee of three — the placements produce distinct outcomes. Formula derivation: selecting r items from n in order. First slot: n choices. Second: n−1. Third: n−2 ... down to (n−r+1) slots. Total = n × (n−1) × ... × (
Combinations: When Order Doesn't Matter
A combination selects r items from n where sequence is irrelevant — a group is a group regardless of the order members are selected. Every combination of r items corresponds to r! permutations of the same items, so we divide nPr by r! to remove redundant orderings. Example: choosing 5 lottery numbers from 1–50. Order doesn't matter (5-10-23-31-42 i
Fundamental Probability Rules
Classical probability: P(A) = (favorable outcomes) / (total equally likely outcomes). All probabilities are between 0 and 1; P(certain event) = 1; P(impossible event) = 0. Complement rule: P(A does not occur) = 1 − P(A). Example: P(not rolling a 6) = 1 − 1/6 = 5/6. Addition rule: P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events (
Frequently Asked Questions
What is the difference between permutations and combinations?
Order matters for permutations; order is irrelevant for combinations. Selecting a president, vice president, and secretary from 10 candidates is a permutation (10P3 = 720) because the roles differ. Selecting any committee of 3 from 10 is a combination (10C3 = 120). A simple test:
What does n! (factorial) mean?
n! (n factorial) = n × (n−1) × (n−2) × … × 2 × 1. Examples: 5! = 120, 6! = 720, 10! = 3,628,800. Special case: 0! = 1 (by definition). Factorials grow extremely rapidly — 20! ≈ 2.4 × 10¹⁸, which is why even small combinatorics problems involve huge numbers.
How do I calculate the probability of winning a lottery?
Most lotteries are combination problems. For Powerball: choose 5 from 1–69 and 1 from 1–26. Jackpot probability = 1/(69C5 × 26) = 1/292,201,338 ≈ 0.00000034%. For a simpler state lottery (pick 6 from 49): 1/(49C6) = 1/13,983,816 ≈ 0.0000072%. The expected value is always negative
What is conditional probability?
Conditional probability P(A|B) is the probability of event A given that event B has already occurred. Formula: P(A|B) = P(A and B) / P(B). Example: P(drawing a second Ace from a deck, given first card was an Ace) = P(second Ace | first was Ace) = 3/51 ≈ 5.9%, not 4/52 = 7.7%, bec
What is the binomial probability formula?
Binomial probability: P(exactly k successes in n trials) = nCk × p^k × (1−p)^(n−k), where p is the probability of success on one trial. Example: flipping a fair coin 10 times, probability of exactly 6 heads: 10C6 × 0.5^6 × 0.5^4 = 210 × 0.015625 × 0.0625 = 0.2051 ≈ 20.5%.