∑(20Ck)^2, k∈(0,20) is equal to : (1) 40C21 (2) 40C19 (3) 40C20 (4) 41C20

Question

\(\displaystyle\sum_{k=0}^{20}\)(²⁰C_k)² is equal to :

∑(²⁰C_k)², k∈(0,20)

(1) ⁴⁰C₂₁

(2) ⁴⁰C₁₉

(3) ⁴⁰C₂₀

(4) ⁴¹C₂₀

Answer 1

Option : (3). ⁴⁰C₂₀

∑(²⁰C_k)², k∈(0,20)

 \(\displaystyle\sum_{k=0}^{20}\)²⁰C_k.²⁰C_20-k

sum of suffix is const. so summation will be ⁴⁰C₂₀.