The value of r = 0 22 C r C r 23 22 is [2023]

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Solution

Q.
The value of $\sum_{r = 0}^{22} {}^{22}{C_{r} {}^{23}{C_{r}}}$ is [2023]

1 ${}^{45}{C_{24}}$

2 ${}^{45}{C_{23}}$

3 ${}^{44}{C_{23}}$

4 ${}^{44}{C_{22}}$

Ans.
(2)

$\sum_{r = 0}^{22} {}^{22}{C_{r}} {}^{23}{C_{r}}$

$= \sum_{r = 0}^{22} {}^{22}{C_{r}} {}^{23}{C_{23 - r}}$

$= {}^{22 + 23}{C_{23}} = {}^{45}{C_{23}}$

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