JEE Main 2019 (Online) 11th January Morning ShiftQuestion 1
The value of $r$ for which ${}^{20}C_r\,{}^{20}C_0+{}^{20}C_{r-1}\,{}^{20}C_1+{}^{20}C_{r-2}\,{}^{20}C_2+\ldots+{}^{20}C_0\,{}^{20}C_r$ is maximum, is :
Binomial Theorem · Properties and Sums of Binomial Coefficients
Solution
Concept. The sum is the coefficient of $x^r$ in $(1+x)^{20}(1+x)^{20}$. Formulas used. (1) Vandermonde: $\sum_k{}^{m}C_{r-k}\,{}^{n}C_k={}^{m+n}C_r$(2) ${}^{40}C_r$ is greatest at $r=20$Step 1 - Simplify. By (1), the sum is ${}^{40}C_r$. Step 2 - Maximum. By (2), $r=20$. Answer: (B)