Chapter 7 — Binomial Theorem

Chapter 7 of the Class 11 Maths NCERT textbook, "Binomial Theorem", provides a formula to expand (a + b)ⁿ for any positive integer n without repeated multiplication, using binomial coefficients ⁿCᵣ from the terms ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + … + ⁿCₙbⁿ.

• Expanding powers efficiently — The chapter shows how the Binomial Theorem replaces tedious repeated multiplication with a single expansion formula for any positive integer power, making even high powers of a binomial quick to write out.
• Where the coefficients come from — It connects the expansion's coefficients to Pascal's triangle and to combinations, revealing that the numbers in each row aren't arbitrary but the same ⁿCᵣ counts studied earlier in the course.
• Proof and practical uses — The theorem is justified by mathematical induction and then applied to real tasks — evaluating large powers, handling special cases like (1 + x)ⁿ, and proving divisibility results — showing it as a working tool, not just a formula.

1. 01Binomial Theorem: (a + b)ⁿ = ΣⁿCₖaⁿ⁻ᵏbᵏ from k=0 to n, where ⁿCᵣ = n!/(r!(n–r)!)
2. 02Pascal's triangle and binomial coefficients provide expansion coefficients without multiplication
3. 03There are always (n+1) terms in (a + b)ⁿ expansion, with indices of a and b summing to n in each term
4. 04Special cases: (x – y)ⁿ uses alternating signs; (1 + x)ⁿ simplifies to ⁿC₀ + ⁿC₁x + ⁿC₂x² + ...
5. 05Practical applications include computing large powers (e.g., (98)⁵ = 9039207968) and proving divisibility (e.g., 6ⁿ – 5ⁿ ≡ 1 mod 25)
6. 06The theorem simplifies via the summation notation and is proven using mathematical induction