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Lesson 1 of 8

Add cases, multiply stages, subtract bad outcomes

Choose the counting operation from the structure of an outcome.

Understand the idea

Add counts for disjoint alternatives. Multiply counts for consecutive stages when each earlier choice has the stated number of continuations. If branch sizes differ, add their products instead of multiplying by a guessed common count. A complement subtracts bad outcomes from a well-defined universe.

Trace a small example

Choosing one of 3 workshops or one of 2 talks gives 5 activities. Choosing one shirt from 3 and one pair of trousers from 2 gives 6 outfits. For three-digit decimal numbers, the first digit has 9 choices while later digits have 10 each: 900 numbers.

Conditions and cost

Distinguish codes from numbers: a code may permit leading zeros. Decide whether symbols may be reused. A complement is valid only if the unwanted objects belong to the same universe and are counted once.

Pause & predict

How many two-digit decimal numbers exist?

Choose an answer
Try it yourself

Turn the idea into code

How many length-3 binary strings contain at least one 1?

Give me a hint

Remove the one string containing only zeros from all strings.

Show the explained solution

There are 2^3 − 1 = 7. Designating one required 1 position would count strings with several 1s more than once.

Your notes

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Mark complete when you can explain it yourself.

From a problem to a programIdentify the input, output, and constraints, then translate a small plan into C++.Variables, types, and remembered valuesChoose a representation, initialize it, and understand what assignment really changes.Arithmetic without surprisesPredict division and remainder, widen before multiplication, and format decimal answers.Conditions and logical decisionsBuild exclusive categories, combine comparisons, and protect operations with short-circuit logic.Optional: other ways to chooseRecognize switch for exact choices and the conditional operator for choosing a value.For loops: make repetition preciseTrace initialization, condition, body, and update—and avoid an extra iteration.While, do-while, and stopping rulesChoose a loop from its stopping condition and read the exact input format.Break, continue, and early answersDistinguish leaving a loop from skipping one iteration, without losing required input.Five patterns worth understandingUse sums, counts, best-so-far candidates, products, and flags to process a stream.Nested loops: rows and columnsTranslate a shape into two loop roles and count the total work.Take a number apart, one digit at a timeUse remainder and division to process decimal digits, including zero.Multiple cases and a debugging mindsetCombine the patterns, isolate each case, and test the boundaries that can break your idea.Arrays: give every value a placeUnderstand indices, initialization, capacity, active length, and the first safe array loop.Scan, count, search, and compareDistinguish values from positions and preserve the exact tie rule a problem asks for.Change a sequence without losing itUpdate, copy, compare, reverse, and inspect neighbours using the correct bounds.Matrices: think in two coordinatesTraverse tables, total rows and columns, inspect diagonals, and protect grid neighbours.Characters become stringsSeparate characters from numbers, then safely inspect and edit managed text.Read the text the problem actually givesChoose words or full lines, preserve empty cases, and reset only the state that needs it.Scan text: count, filter, transformBuild a result from matching characters without losing leading zeros or skipping shifted text.Join, compare, and find textLearn content comparison, lexicographical order, substring counts, and the not-found sentinel.