The binomial theorem is one of the most powerful tools in algebra. It helps expand expressions like (a+b)n(a + b)^n(a+b)n without multiplying them repeatedly. Whether you’re solving expansion questions, probability problems, or higher-level mathematics, this formula saves time and simplifies complex expressions. Students preparing for CUET, SSC, JEE, Banking, or any competitive exam often rely on it to handle polynomial expansions quickly and accurately.
Formula for Calculating Binomial Theorem – Overview
| Formula | Variables & Meaning | When it is Used |
|---|---|---|
| (a + b)ⁿ = Σ [ nCr × aⁿ⁻ʳ × bʳ ] | n = exponent, r = term index, a & b = variables | Used to expand powers, solve algebraic expressions, and in probability |
What is Binomial Theorem in Maths?
In mathematics, the binomial theorem is used to expand expressions of the form (a+b)n(a + b)^n(a+b)n into a sum of terms. Each of these terms contains coefficients (nCr), powers of aaa, and powers of bbb. Instead of expanding manually through repeated multiplication, the binomial theorem provides a direct formula.
The term structure follows a clear pattern:
- The first term is always ana^nan.
- The last term is always bnb^nbn.
- The middle terms combine different powers of aaa and bbb.
The binomial theorem is essential in algebra, probability theory (binomial distribution), and higher mathematics. It is commonly tested in exams like CUET, JEE, SSC, Banking, and other competitive tests.
Examples to Calculate Binomial Theorem
Example 1: Expand (a + b)² using the binomial theorem
Step 1: (a + b)² = ²C₀a² + ²C₁ab + ²C₂b²
Step 2: = (1)(a²) + (2)(ab) + (1)(b²)
Result: (a + b)² = a² + 2ab + b²
Example 2: Expand (x + 2)³ using the binomial theorem
Step 1: (x + 2)³ = ³C₀x³ + ³C₁x²(2) + ³C₂x(2²) + ³C₃(2³)
Step 2: = (1)(x³) + (3)(2x²) + (3)(4x) + (1)(8)
Step 3: = x³ + 6x² + 12x + 8
Result: (x + 2)³ = x³ + 6x² + 12x + 8
FAQs about Binomial Theorem Formula
Q1. What is the binomial theorem used for?
It is used to expand expressions like (a+b)n(a + b)^n(a+b)n, in probability (binomial distribution), and in algebra.
Q2. What are binomial coefficients?
They are values of nCr = n! / (r!(n – r)!) and appear as coefficients in expansion.
Q3. Is (a – b)ⁿ expanded the same way?
Yes, but terms with odd powers of b become negative due to the minus sign.
Q4. Why is binomial theorem important in exams?
It appears frequently in CUET, SSC, Banking, and JEE, especially for expansion and simplification.
Q5. What is the middle term in a binomial expansion?
If n is even → the middle term is the (n/2+1)(n/2 + 1)(n/2+1)th term.
If n is odd → there are two middle terms.
Q6. Can binomial theorem be applied to negative powers?
Yes, with modifications. This extended version is known as the generalized binomial theorem.