If x, y and r are positive integers, then ^x{ | vedclass

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(C) The given expression is $^x{C_r} \cdot ^y{C_0} + ^x{C_{r-1}} \cdot ^y{C_1} + ^x{C_{r-2}} \cdot ^y{C_2} + \dots + ^x{C_0} \cdot ^y{C_r}$.This is a

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$^{x+y}C_r$

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(C) The given expression is $^x{C_r} \cdot ^y{C_0} + ^x{C_{r-1}} \cdot ^y{C_1} + ^x{C_{r-2}} \cdot ^y{C_2} + \dots + ^x{C_0} \cdot ^y{C_r}$.This is a standard identity known as Vandermonde's Identity.According to Vandermonde's Identity, the sum of the products of combinations $\sum_{k=0}^{r} {^x{C_{r-k}} \cdot ^y{C_k}}$ is equal to $^{x+y}C_r$.This represents the number of ways to choose $r$ items from a total of $x+y$ items (where $x$ items are of one type and $y$ items are of another type).Therefore, the correct option is $C$.

If $x, y$ and $r$ are positive integers, then $^x{C_r} + ^x{C_{r-1}} \cdot ^y{C_1} + ^x{C_{r-2}} \cdot ^y{C_2} + \dots + ^y{C_r} = $

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