Binomial coefficient
- Olympiad · IOQM
A binomial coefficient, written ⁿCᵣ, is the number of ways to choose r things from n different things when order does not matter.
It is read as n choose r. The formula is ⁿCᵣ = n! / (r! × (n - r)!). Here n! means 1 × 2 × 3 × ... × n. So 5! = 120 and 0! is taken as 1.
The same numbers appear when you open up (a + b)ⁿ. That is why they are called binomial coefficients. For n = 3, (a + b)³ = a³ + 3a²b + 3ab² + b³. The numbers 1, 3, 3, 1 are ³C₀, ³C₁, ³C₂ and ³C₃.
A small example
In how many ways can you pick 2 friends out of 5 for a team?
- ⁵C₂ = 5! / (2! × 3!) = 120 / (2 × 6) = 10.
- Check by listing, with friends A to E: AB, AC, AD, AE, BC, BD, BE, CD, CE, DE. That is 10.
One useful property: ⁿCᵣ = ⁿCₙ₋ᵣ. Choosing 2 friends to take is the same as choosing 3 to leave out. So ⁵C₂ = ⁵C₃ = 10. Another: each number in Pascal's triangle is the sum of the two above it. For counting where order does matter, start from the multiplication principle.
Where it fits
This goes beyond the Class 10 board syllabus. HBCSE's syllabus for the Mathematical Olympiad names it under algebra. 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 (Algebra) · HBCSE, TIFR
- Mathematical Olympiad 2026-2027: stages of selection · HBCSE, TIFR
- Brochure: Mathematical Olympiads 2026-2027 · HBCSE, TIFR
About the author
Chief Architect
An engineer (B.Tech, Computer Science) with over 20 years of preparing students for entrance exams, including as Dean at FIITJEE and Vice-President at Aakash. He is the chief architect of the TRUpreBoards evaluation engine.
Mohit Sardana on LinkedIn