Binomial Theorem
Vandermonde's Identity
nta_pyq_2023_jan
Grade None
Question:
The value of $\displaystyle\sum_{r=0}^{22}\,{}^{22}C_r\cdot{}^{23}C_r$ is:
${}^{45}C_{23}$
${}^{44}C_{23}$
${}^{45}C_{24}$
${}^{44}C_{22}$
Step-by-Step Solution
Key Concept: Use ${}^{23}C_r={}^{23}C_{23-r}$. Then $\sum_{r=0}^{22}{}^{22}C_r\cdot{}^{23}C_{23-r}={}^{45}C_{23}$ by Vandermonde's identity.
$\displaystyle\sum_{r=0}^{22}{}^{22}C_r\cdot{}^{23}C_{23-r}={}^{45}C_{23}$.
Correct Answer: 1