Combinatorics and Probability
Core concepts
Counting and probability questions test the multiplication principle, permutations versus combinations, factorials, arrangements with restrictions, and basic probability rules. Know that P(A or B) = P(A) + P(B) - P(A and B), that P(A and B) = P(A)P(B) for independent events, and that conditional probability is P(A|B) = P(A and B)/P(B). Distinguish ordered selections (permutations, nPr) from unordered ones (combinations, nCr). Complementary counting - counting the opposite and subtracting from the total - is often the fastest route.
Traps to watch
- 'At least one' is almost always easier via the complement.
- Circular arrangements fix one item, giving (n-1)! orders.
- Repeated items divide the permutation count by the factorial of each repetition count.
- With or without replacement changes the denominator.
- Mutually exclusive is not the same as independent.
Rule of thumb
Ask whether order matters. If yes, use permutations; if no, use combinations. If the question says 'at least', use the complement.
Worked example
A bag has 4 red and 6 blue balls. Two are drawn without replacement. What is the probability both are red? P = (4/10)(3/9) = 12/90 = 2/15. Alternatively, C(4,2)/C(10,2) = 6/45 = 2/15. For 'at least one red', compute 1 - P(no red) = 1 - C(6,2)/C(10,2) = 1 - 15/45 = 2/3.