Answer
(i) The permutations of five objects taken 2 at a time without repetition are:
$\begin{array}{ccc}
\\a, b &b, a
\\a, c &c,a
\\a, d &d,a
\\a, e &e,a
\\b, c &c, b
\\b, d &d, b
\\b, e &e,b
\\c, d &d, c
\\c, e &e, c
\\d, e &e, d
\end{array}$
(ii) $_5P_2=20$
Work Step by Step
List all permutations of the five objects taken two at a time without repetition to have:
a, b
a, c
a, d
a, e
b, c
b, d,
b, e
c, d
c, e
d, e
b, a
c, a
d, a
e, a
c, b
d, b
e, b
d, c
e,c
e, d
There are 20 permutations.
Evalue $_5P_2$ to have:
$\require{cancel}_5P_2= \dfrac{5!}{(5-2)!}
\\_5P_2= \dfrac{5!}{3!}
\\_5P_2= \dfrac{5(4)(3!)}{3!}
\\_5P_2= \dfrac{5(4)\cancel{(3!)}}{\cancel{3!}}
\\_5P_2=20$
