Explanation

(a)
Given that, $\ \ \ \ ^{n + 2}C_{8}:\ ^{n - 2}P_{4} = \frac{57}{16}$
$\Rightarrow \ \ \ \ \frac{(n + 2)(n + 1)n(n - 1)}{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8} = \frac{57}{16}$
$\Rightarrow \ \ \ (n + 2)(n + 1)n(n - 1)$
$= \frac{57}{16} \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8$
$= 57 \times 3 \times 4 \times 5 \times 6 \times 7$
$= 21 \times 20 \times 19 \times 18$
$\Rightarrow \ \ n = 19$