Binomial Theorem – Class 11 Maths (NCERT Based)
The chapter Binomial Theorem is an important part of Class 11 Mathematics. It provides a shortcut method to expand expressions of the form (a + b)ⁿ without multiplying repeatedly. This theorem is widely used in algebra, probability, and combinatorics.
This guide strictly follows the NCERT Class 11 Maths syllabus and explains concepts in a clear, student-friendly way.
📖 What is the Binomial Theorem?
For a positive integer n, the binomial theorem states:
(a + b)ⁿ = Σ (nCr × a^(n−r) × b^r),
where r = 0, 1, 2, …, n
Here:
- nCr = n! / [r! × (n − r)!] (Binomial Coefficient)
- a^(n−r) × b^r represents the terms in the expansion
🔹 Binomial Coefficients
The coefficients nCr have important properties:
- Symmetry: nCr = nC(n−r)
- Sum of coefficients: 2ⁿ = Σ nCr
- Pascal’s Triangle: Each row represents coefficients of (a + b)ⁿ
🔹 General Term
The (r + 1)th term in the expansion of (a + b)ⁿ is:
T(r + 1) = nCr × a^(n−r) × b^r
This is used to find specific terms in the expansion without writing the entire expression.
🔹 Middle Term
- If n is even, middle term = T(n/2 + 1)
- If n is odd, there are two middle terms: T((n+1)/2) and T((n+3)/2)
🔹 Important Special Cases
- (1 + x)ⁿ → Expansion gives powers of x with coefficients nCr
- (a − b)ⁿ → Alternate signs appear due to negative b: (−1)^r factor
🔹 Properties of Binomial Coefficients
- nC0 = 1, nCn = 1
- Σ nCr = 2ⁿ (sum of all coefficients)
- Σ (−1)^r × nCr = 0