LOGIC & PUZZLES / COMBINATORICS
Count choices
without counting twice.
The key question is often not “which formula?” but “when do I consider two outcomes the same?”
THE MAIN IDEA
Order and repetition change what is being counted.
The multiplication principle says that if one choice can be made in m ways and a later independent choice can be made in n ways, there are m times n ordered combinations of those choices. A three-digit code with ten possibilities at each place has one thousand sequences if repeats are permitted. If repeats are forbidden, the number of options shrinks after each position. A different question—choosing a group of three people for an unranked team—does not treat the six possible orderings of a particular trio as six different teams. These distinctions separate permutations, where order matters, from combinations, where the chosen set is the same in any order. Listing small cases before applying a formula is a reliable antidote to memorizing the wrong expression. Draw a decision tree for two or three choices and check how many leaves it has. If the tree becomes large, identify repeated structures and use factorials or binomial coefficients. Watch for constraints that overlap: “wears a hat” and “has a red shirt” categories can both include one person, so adding category totals may double-count the intersection. Inclusion-exclusion corrects this by subtracting the overlap once, then adding triple intersections back as appropriate. Another common source of error is forgetting that an event has zero possibilities after an earlier choice rules them out. Define the experiment before counting: are seats distinct, can an object be reused, do rotations count as the same arrangement, and are participants distinguishable? Circular arrangements often identify rotations differently from ordinary lineups. If the rules are ambiguous, write both interpretations and calculate them separately rather than quietly picking one. The result is only as meaningful as the sample space: a neat formula cannot repair a model that counted outcomes incorrectly. In games, choosing a random hand is different from choosing a random sequence of draws, even if the final objects look the same, because the probabilities may differ. For difficult cases, create a small test instance, enumerate every possibility, and compare that list with the proposed formula. Counting gives the foundations for probability, scheduling, password-space estimates, and algorithm analysis. Accuracy begins with a plain-language definition of “one outcome,” and ends with a check that every legal outcome appears exactly once.
TRY THIS
Count a tiny version by hand.
Choose three colored cards and two positions. List ordered selections when repeats are not allowed, then regroup them into unordered pairs. Compare the counts and explain why one arrangement may appear twice in the ordered list.