The value of ∑r=022 22Cr23Cr is

∑r=022 22Cr⋅23Cr=∑r=022 22Cr⋅23C(23−r)=22C0⋅23C23+22C1⋅23C22+…+22C22⋅23C1= 45C23 (using Vander Mond’s theorem)

The value of ∑r=022 22Cr23Cr is

∑r=022 22Cr⋅23Cr=∑r=022 22Cr⋅23C(23−r)=22C0⋅23C23+22C1⋅23C22+…+22C22⋅23C1= 45C23 (using Vander Mond’s theorem)

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

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$^{45} C_{24}$

$^{45} C_{23}$

$^{44} C_{23}$

$^{44} C_{22}$

answer is B.

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Detailed Solution

$\begin{array}{l} \sum_{r = 0}^{22}^{22} C_{r} \cdot^{23} C_{r} = \sum_{r = 0}^{22}^{22} C_{r} \cdot^{23} C_{(23 - r)} \\ =^{22} C_{0} \cdot^{23} C_{23} +^{22} C_{1} \cdot^{23} C_{22} + \dots +^{22} C_{22} \cdot^{23} C_{1} \end{array}$
= $^{45} C_{23}$ (using Vander Mond’s theorem)