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Question

Mathematics Question on Combinations

The value of ∑r=02222Cr23Cr\displaystyle\sum_{r=0}^{22}{ }^{22} C_r{ }^{23} C_r is

A

45C24{ }^{45} C _{24}

B

45C23{ }^{45} C_{23}

C

44C23{ }^{44} C _{23}

D

44C22{ }^{44} C_{22}

Answer

45C23{ }^{45} C_{23}

Explanation

Solution

∑r=022  22Cr.23Cr\overset{22}{\underset{r=0}\sum} {\,\,} ^{22}C_r. ^{23}C_r

=∑r=022  22Cr.23C23−r=\overset{22}{\underset{r=0}\sum} {\,\,} ^{22}C_r. ^{23}C_{23-r}

=  45C23={\,\,}^{45}C_{23}

Hence, the correct option is (B): 45C23^{45}C_{23}