Explanation

(b)
Given, $\ ^{n}C_{r} = 30240$ and $\ ^{n}C_{r} = 252$
$\frac{n!}{(n - r)!} = 30240$ and $\frac{n!}{(n - r)!r!} = 252$
$\Rightarrow \ \ \ r! = \frac{30240}{252} = 120\ \ \ \ \Rightarrow \ \ r = 5$
$\therefore\ \ \ \ \frac{n!}{(n - 5)!} = 30240$
$\Rightarrow \ \ \ \ \ n(n - 1)(n - 2)(n - 3)(n - 4)$
$= 10(10 - 1)(10 - 2)(10 - 3)(10 - 4)$
$\Rightarrow \ \ \ \ n = 10$
Hence, required ordered pair is (10, 5)