Counting questions ask how many arrangements, committees or codes exist; probability questions ask how likely an outcome is. Both reduce to the same two ideas — multiply when choices happen in sequence, divide when you compare a part with a whole — and both are heavily represented because they are quick to mark and hard to fake.
The single highest-yield trick in this whole family is the complement: "at least one" is almost always easier as "one minus none". Questions that take three minutes by listing take twenty seconds by complement.
Counting: multiply along a sequence of choices, add across alternatives. Probability: favourable outcomes divided by total outcomes, where both are counted the same way.
The complementary rule deserves its own line because it is a technique, not a definition: whenever a question says at least one, ask what "none" looks like, because "none" is a single product while "at least one" is many cases.
If a task has m ways for its first stage and n ways for its second, the pair has m × n ways. Three shirts and four trousers give 12 outfits; five digits with no repetition give 5 × 4 × 3 = 60 three-digit codes.
| Situation | Formula | Example |
|---|---|---|
| Arranging all of n objects | n! | 5 books: 120 orders |
| Choosing r from n, order matters | n! ÷ (n − r)! | President and secretary from 5: 5 × 4 = 20 |
| Choosing r from n, order irrelevant | n! ÷ (r! (n − r)!) | Committee of 2 from 5: 10 |
Values worth knowing cold: 4! = 24, 5! = 120, 6! = 720; selecting 2 from 5 gives 10, 2 from 10 gives 45, 3 from 6 gives 20, 3 from 7 gives 35. The dividing line is the word arrange or order: if different orders count as different outcomes, use permutations; if not, combinations.
Order is irrelevant, so use combinations: 7 × 6 × 5 ÷ (3 × 2 × 1) = 210 ÷ 6 = 35.
Total outcomes 6 × 6 = 36. Favourable pairs: (2,6), (3,5), (4,4), (5,3), (6,2) — five of them. Probability = 5/36, which is about 0.139.
(4/10) × (3/9) = 12/90 = 2/15. The second fraction uses 3/9 because one red ball has left the bag: forgetting that gives the wrong answer 16/100.
Complement: the only case with no head is TTTT, whose probability is (1/2)⁴ = 1/16. So the answer is 1 − 1/16 = 15/16. Listing the fifteen favourable cases would take a minute; the complement takes ten seconds.
A three-digit code is formed from 1 to 5 with no repetition. How many codes are possible?
A card is drawn from a standard pack. What is the probability that it is a heart or a king?
A fair coin is tossed twice. What is the probability of at least one tail?
A box holds 3 defective and 7 good items. Two are chosen at random without replacement. What is the probability that both are good?
Without looking back: when do you multiply probabilities, and how do you handle "at least one"?
The answer: multiply for independent events happening together; use 1 minus the probability of none for "at least one".
Probabilities of independent events are . To find the probability of at least one success, subtract the probability of from 1. When selecting without replacement, the denominator of the second draw . Choosing a committee ignores order, so it uses combinations rather than .
A question asks: "In how many ways can 2 boys and 2 girls be chosen from 4 boys and 3 girls?" Solve it, then say what changes if the question instead asks for the probability that a random committee of 4 contains exactly 2 girls.
Choosing is unordered, so multiply the two combinations: 2 boys from 4 is 6 ways (4 × 3 ÷ 2), and 2 girls from 3 is 3 ways, giving 18 committees.
For the probability version, count the total committees of 4 from 7 people: 7 × 6 × 5 × 4 ÷ 24 = 35. The probability is 18/35, which is about 0.514 — just over a half.
The lesson in the transformation is that counting and probability are the same skill applied twice: probability is a favourable count over a total count, so once the counting is right the probability is one division away.
Counting and probability close the first pass over quantitative reasoning. The next lessons add the algebraic tools that many questions hide behind: powers and roots, then linear algebra and quadratics. The timed mixed drill comes later, once every technique is in place.