Σₖ₌₀²⁰ ( ²⁰Cₖ )² is equal to :

Solution

Maths sir · 3 steps

Write one square as a product of two coefficients, then read it as a coefficient of (1+x)⁴⁰.

1. Use symmetry. Since ²⁰Cₖ = ²⁰C₂₀₋ₖ, each square can be written as ²⁰Cₖ × ²⁰C₂₀₋ₖ. The two lower indices now add up to 20.
2. Multiply two expansions. In (1+x)²⁰(1+x)²⁰, the term x²⁰ comes from picking xᵏ in the first bracket and x²⁰⁻ᵏ in the second. Adding over all k gives the coefficient of x²⁰ as Σₖ₌₀²⁰ ²⁰Cₖ ²⁰C₂₀₋ₖ.

   Σₖ₌₀²⁰ (²⁰Cₖ)² = coefficient of x²⁰ in (1+x)⁴⁰ = ⁴⁰C₂₀

3. So. The sum equals ⁴⁰C₂₀, which is option (c).

The idea to keep. A sum of products of coefficients whose lower indices add to a constant is one coefficient of a product of two binomials. In general, Σ (ⁿCₖ)² = ²ⁿCₙ.

Answer from the NTA final answer key. Solution by PrepWiser.