Explanation

(b)
$\because\ \ \ \ \ \ \ ^{n}P_{r} = 1680$
$\Rightarrow \ \ \ \frac{n!}{(n - r)!} = 1680$ …(i)
And $\ ^{n}C_{r} = 70$
$\Rightarrow \ \ \ \frac{n!}{(n - r)!r!} = 70$ …(ii)
From Eqs. (i) and (ii), we get
$r! = \frac{1680}{70} = 24 = 4!\ \ \ \Rightarrow \ \ \ r = 4$
On putting the value of $r$ in Eq. (i), we get
$\Rightarrow \ \ \ \frac{n!}{(n - 4)!} = 1680$
$\Rightarrow \ \ \ n(n - 1)(n - 2)(n - 3) = 1680$
$\Rightarrow \ \ \ n(n - 1)(n - 2)(n - 3) = 8(8 - 1)(8 - 2)(8 - 3)$
$\Rightarrow \ \ \ n = 8$
$\therefore\ \ \ \ 69n + r! = 69 \times 8 + 24 = 576$