Sign in

Principle deep dive

Inclusion-Exclusion (sets + probability)

n(A∪B) = n(A) + n(B) − n(A∩B) and the three-set generalisation. The complement form 1 − P(none) is the other face of the same identity, used heavily in Binomial Distribution and Probability via Counting questions.

questions in the bank
44
tagged HARD
14%
chapter spread
3
worked examples below
4

When to reach for it

Count or probability of "at least one", "exactly two", or a union of overlapping events / sets.

Why this principle matters

Two formulas cover ~95% of NDA's inclusion-exclusion questions. For two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). For three: n(A ∪ B ∪ C) = sum of singles − sum of pairs + n(A ∩ B ∩ C). The probability versions are identical with n → P.

The challenge isn't the formula — it's parsing the question. 'Exactly two' means double-counted in the union but subtracted in the triple intersection. 'At least one' is the easier flip side: P(at least one) = 1 − P(none) — almost always faster than the inclusion-exclusion expansion.

Set and probability framings are the same principle. A 'class of 45 students, cricket and football' question is identical to 'two events, classical probability' — only the words change. Recognising this saves you re-learning the trick per chapter.

4 worked examples from the bank

Each example demonstrates the principle on a real past-year question. Click to reveal the answer, then the solution.

Example 1Sets & RelationsEASY
In a class of 45 students, 34 like to play cricket and 26 like to play football. Further, each student likes to play at least one of the two games. How many students like to play exactly one game?

[Q11 · Sep · 2025]

Example 2Sets & RelationsMODERATE
Suppose set AA consists of first 250 natural numbers that are multiples of 3 and set BB consists of first 200 even natural numbers. How many elements does A∪BA\cup B have?

[Q43 · Sep · 2021]

Example 3ProbabilityMODERATE
The probability that a student passes Physics test is 2/3 and the probability that he passes both Physics test and English test is 11/15. The probability that he passes at least one test is 4/5. What is the probability that he passes English test?

[Q108 · Sep · 2025]

Example 4Sets & RelationsHARD
In a class of 240 students, 180 passed in English, 130 passed in Hindi and 150 passed in Sanskrit. Further, 60 passed in only one subject, 110 passed in only two subjects and 10 passed in none of the subjects. How many passed in all three subjects?

[Q40 · Sep · 2024]

Variants to recognise

Same principle, different surfaces. Pattern-match these on test day.

  • Two-set: |A ∪ B| = |A| + |B| − |A ∩ B|

    The base. The −|A∩B| corrects the double count. Identical for probabilities.

  • Three-set: + singles − pairs + triple

    |A ∪ B ∪ C| = ΣAᵢ − Σ(Aᵢ ∩ Aⱼ) + |A ∩ B ∩ C|. Sign alternates by subset size.

  • "Exactly one" / "exactly two"

    Exactly-k counts: sum over k-fold intersections with alternating signs. Easier to draw the Venn and read off regions.

  • P(at least one) = 1 − P(none)

    Complement trick. Whenever the question says 'at least one X', compute P(none of them) and subtract. Saves an inclusion-exclusion expansion.

Drill every inclusion-exclusion (sets + probability) question

44 questions from the bank — paginated, with cart and Word-export support.

Drill the 44 questions

Related principles

Often combined with this one — drill these next if you found the examples above tractable.