Summation simplification!!

Needed a simplification of this result:- $\frac{\sum_{r=1}^{n}({n-1\choose r-1}\sum_{i=0}^{r-1}[(-1)^{i}{r\choose i}{n(r-i)\choose n}])}{\sum_{r=1}^{n}({n\choose r}\sum_{i=0}^{r-1}[(-1)^{i}{r\choose i}{n(r-i)\choose n}])}$ Here, $\sum_{i=0}^{r-1}[(-1)^{i}{r\choose i}{n(r-i)\choose n}]$ is also the coefficient of $x^{n}$ in the expansion of $((1+x)^{n}-1)^{r}$ .

Background (if anyone's interested):-

Ended up at this trying to solve a problem I randomly created, which is as follows:-

'There are $n$ unique sets with $n$ unique items in each. If exactly $n$ items are selected from all of these, find the probability of finding a set with at least 1 item selected in it'.

Also, if this attains a specific value as $n→∞$ , what is that value?

#Combinatorics

Easy Math Editor

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Comments

Probability of a set being chosen and having none of the selected items is $\dfrac{^{n^2-n}C_n}{^{n^2}C_n}$ , so the req prob is $1-\dfrac{^{n^2-n}C_n}{^{n^2}C_n}$

Jason Gomez - 5 days, 18 hours ago

Thanks. I ended up using an explicit approach and reached a dead end. Didn't think of this :).

Deepankur Jain - 5 days, 6 hours ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$