Learning pathOriginal reference
Lesson 6 of 8

Count contributions and correct overlap

Group equal pairs and derive inclusion-exclusion by tracking how often one object is counted.

Understand the idea

If a value occurs f times, it contributes C(f, 2) unordered equal-value index pairs. Add over values. For two sets, |A ∪ B| = |A| + |B| − |A ∩ B| because intersection members were counted twice.

Trace a small example

Frequencies 3 and 2 contribute 3 + 1 = 4 equal pairs. For three sets, add singleton counts, subtract pair intersections, then add the triple intersection back: a member in all three was counted 3 − 3 + 1 = 1 time.

Conditions and cost

Intersections include members also present in further sets. Use wide arithmetic for f(f − 1)/2. A frequency scan is often faster and clearer than checking O(n²) pairs individually.

Frequency array

The value becomes an index. Each arrival increases freq[value] by one. This domain has ten possible values.

Input values

20
11
22
33
24
55

freq[value] — indices are values

00
11
32
13
04
15
06
07
08
09

The frequencies add up to 6, the number of input values.

Guided simulation · Change the inputs to test your understanding.

Pause & predict

Why add the triple intersection back in three-set inclusion-exclusion?

Choose an answer
Try it yourself

Turn the idea into code

Among 30 students, 18 know C++, 12 know Python, and 7 know both. How many know neither?

Give me a hint

Find the union before subtracting it from the class.

Show the explained solution

The union has 18 + 12 − 7 = 23 students, so 30 − 23 = 7 know neither.

Your notes

Ready for the next idea?

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.