If the number of selections of r as well as (n+r) things from 5n...

The number of selections of $r$ things from $5n$ different things is given by ^{5n}C_{r}$.<br><br>The number of selections of$(n+r)$things from 5n$ different things is given by ^{5n}C_{n+r}$.<br><br>Given that ^{5n}C_{r} = ^{5n}C_{n+r}$.

Using the property of combinations, if ^{N}C_{x} = ^{N}C_{y}$, then either$x = y$or$x + y = N$.<br><br>Since$r \neq n+r$(for$n \neq 0$), we must have:<br><br>$r + (n+r) = 5n$<br><br>2r + n = 5n$

$2r = 4n$

$r = 2n$

Answer: $2n$