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Middle Terms

∑r=02020r3025...

Question

$20 \sum r = 0 (\frac{20}{r}) (\frac{30}{25 - r}) =^{x} C_{y}$ , then find $x - y$ .

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Solution

Consider the following series.
$^{m} C_{0}^{n} C_{r} +^{m} C_{1}^{n} C_{r - 1} + . . .^{m} C_{r}^{n} C_{0}$
$=$ Coefficient of $x^{r}$ in $(1 + x)^{m} . (x + 1)^{n}$
$=$ Coefficient of $x^{r}$ in $(1 + x)^{m + n}$
$=^{m + n} C_{r}$
In the above case, $r = 25 - r^{'} + r^{'} = 25$ and $m = 20$ and $n = 30$
Hence
$^{20 + 30} C_{25}$ $=^{50} C_{25}$
Hence $50 - 25$
$= 25$ .

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