Inclusion-exclusion principle
- Olympiad · IOQM
The inclusion-exclusion principle counts the things in two groups by adding both groups, then taking away the ones counted twice.
Say group A and group B share some members. If you add the size of A to the size of B, the shared members are counted twice. So you take them away once. In symbols:
- n(A or B) = n(A) + n(B) - n(A and B).
For three groups the pattern goes on. Add all three sizes. Take away the three overlaps of two groups. Then add back the part common to all three. The name says it: include, exclude, include again.
A small example
How many whole numbers from 1 to 100 are divisible by 2 or by 3?
- Divisible by 2: 100 ÷ 2 = 50 numbers.
- Divisible by 3: 99 is the last one, so 33 numbers.
- Divisible by both means divisible by 6: 96 is the last one, so 16 numbers.
- Answer: 50 + 33 - 16 = 67.
So 100 - 67 = 33 numbers are divisible by neither 2 nor 3. The overlap is found with the LCM of 2 and 3, which is 6. This is a use of HCF and LCM outside the textbook.
Where it fits
This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under combinatorics, as basic enumeration. The Mathematics Teachers' Association (India), MTA(I), conducts IOQM, the first stage of the Mathematical Olympiad Programme that HBCSE organises for NBHM.
Sources
Facts last checked against these sources on 30 September 2026.
- Syllabus for Mathematical Olympiad (Combinatorics) · HBCSE, TIFR
- Mathematical Olympiad 2026-2027: stages of selection · HBCSE, TIFR
- Brochure: Mathematical Olympiads 2026-2027 · HBCSE, TIFR
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