JEE Main 2021 (Online) 17th March Evening ShiftQuestion 1
The value of $\sum_{r=0}^{6}\left({}^{6}C_{r}\cdot{}^{6}C_{6-r}\right)$ is equal to :
Binomial Theorem · Properties and Sums of Binomial Coefficients
Solution
Concept. $\sum{}^{6}C_r\,{}^{6}C_{6-r}$ is the coefficient of $x^6$ in $(1+x)^6(1+x)^6$. Formulas used. (1) Vandermonde: $\sum_r{}^{m}C_r\,{}^{n}C_{k-r}={}^{m+n}C_k$Step 1 - Apply (1). With $m=n=k=6$: $$\sum_{r=0}^{6}{}^{6}C_r\,{}^{6}C_{6-r}={}^{12}C_6=924.$$ Answer: (A)