Binomial Theorem and Its Simple Applications – JEE Mains Mathematics

• by Zenith LMS
• Updated April 2025

1. Binomial Theorem for Positive Integral Index

• The binomial expansion of (a + b)ⁿ for positive integer n is given by:

• (a + b)^n = Σ (r = 0 to n) [C(n, r) · a^(n−r) · b^r]

• Each term is called a binomial term and is of the form:

• C(n, r) · a^(n−r) · b^r

• Where:

• C(n, r) = n! / [r!(n − r)!]

2. General Term of Binomial Expansion

• The general term (T_r+1) of (a + b)ⁿ is:

• T(r+1) = C(n, r) · a^(n−r) · b^r

• Used to find any specific term in the expansion without writing all terms.
• Example:

• 5th term in (2 + x)^6 → T(5) = C(6, 4) · 2^2 · x^4

3. Middle Term(s)

• If n is even:

• Middle term = T((n/2) + 1)

• If n is odd:

• Middle terms = T((n+1)/2) and T((n+3)/2)

4. Simple Applications

• Finding a particular term from the binomial expansion
• Finding the coefficient of a specific term (e.g., x⁵)
• Solving algebraic identities and symmetric expressions
• Approximations using first few terms of binomial expansion:

• (1 + x)^n ≈ 1 + nx + n(n−1)x²/2! + ...

• Useful in probability, algebra, and inequalities problems in JEE