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

We need to find the value of $$\sum_{r=0}^{22} \binom{22}{r} \cdot \binom{23}{r}$$.

Using the identity $$\binom{n}{r} = \binom{n}{n-r}$$, we write $$\binom{22}{r} = \binom{22}{22-r}$$.

So the sum becomes $$\sum_{r=0}^{22} \binom{22}{22-r} \cdot \binom{23}{r}$$.

By Vandermonde's identity: $$\sum_{r=0}^{k} \binom{m}{k-r}\binom{n}{r} = \binom{m+n}{k}$$.

Here $$m = 22$$, $$n = 23$$, $$k = 22$$.

$$\binom{45}{22} = \binom{45}{23}$$, which matches Option A.

The correct answer is Option A: $$\binom{45}{23}$$.