$\underline {\bf{Answer}}:\bf B$
$\underline{\bf{Explanation:}}$
An ascent at any position $\bf{i < n}$ is given for the value which follows $\bf{i}$ is larger than the current value.
$\underline{\bf{Example:}}$
Number of ascents for $(1, 2, 3, 4)$ are $:-$
$ ({1, 2}), ({2, 3}), ({3, 4})$
Now, coming to the question, a permutation is obtained by taking $\bf{i}$ elements for the first $\bf{i}$ positions (which are sorted in descending order) & then taking a permutation of the remaining $(\bf{n-i})$ elements.
$\therefore \mathrm {|P_i| = \binom{n}{i}\ .(n-i)! = \frac{n!}{i}}$
$\therefore$ Answer$\mathrm{ = |P_k|- |P_{k+1}| = \frac{n!}{k!} -\frac {n!}{(k+1)!}}$
$\therefore \bf B$ is the correct option.
Further Reading: https://en.wikipedia.org/wiki/Permutation#Ascents,_descents,_runs_and_excedances