Explanation

(c)
$\frac{\ ^{56}P_{r + 6}}{\ ^{54}P_{r + 3}} = 30800$
$\Rightarrow \ \ \ \ \frac{56!}{54!} \times \frac{(51 - r)!}{(50 - r)!} = 30800$
$\Rightarrow \ \ \ \ \ 56 \times 55 \times (51 - r) = 56 \times 55 \times 10$
$\Rightarrow \ \ \ \ \ \ 51 - r = 10$
$\Rightarrow \ \ \ \ \ \ r = 41$