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The value of $$\sum_{r=0}^{22} {^{22}C_r} \cdot {^{23}C_r}$$ is
Use the identity:
$$\sum_{r=0}^n\binom{n}{r}\binom{m}{r}=\binom{n+m}{n}$$
Here (n=22), (m=23). So,
$$\sum_{r=0}^{22}\binom{22}{r}\binom{23}{r}=\binom{45}{22}$$
$$\binom{45}{22}$$
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