Question

The value of $\displaystyle\sum_{r=0}^{22}{ }^{22} C_r{ }^{23} C_r$ is

• A. ${ }^{45} C _{24}$
• B. ${ }^{45} C_{23}$
• C. ${ }^{44} C _{23}$
• D. ${ }^{44} C_{22}$

Answer 1

The Correct Option is B

\(\overset{22}{\underset{r=0}\sum} {\,\,} ^{22}C_r. ^{23}C_r \)

\(=\overset{22}{\underset{r=0}\sum} {\,\,} ^{22}C_r. ^{23}C_{23-r} \)

\(={\,\,}^{45}C_{23} \)

Hence, the correct option is (B): \(^{45}C_{23} \)

Answer 2

We are tasked with evaluating: \[ \sum_{r=0}^{22} \binom{22}{r} \binom{23}{r}. \] Using the standard identity: \[ \sum_{r=0}^{n} \binom{a}{r} \binom{b}{n-r} = \binom{a+b}{n}, \] we rewrite the given summation: \[ \sum_{r=0}^{22} \binom{22}{r} \binom{23}{r} = \binom{22+23}{23}. \] Simplify: \[ \binom{22+23}{23} = \binom{45}{23}. \] Thus, the value is: \[ \boxed{\binom{45}{23}}. \]