Answer
(i) The permutations of four objects $a, b, c, d$ taken 2 at a time without repetition are:
$\begin{array}{ccc}
\\a, b &b, a
\\a, c &c,a
\\a, d &d,a
\\b, c &c, b
\\b, d &d, b
\\c, d &d, c
\end{array}$
(ii) $_4P_2=12$
Work Step by Step
List the permutations of the four objects taken 2 at a time without repetition to have:
a, b
a, c
a, d
b, c
b, d
c, d
b, a
c, a
d, a
c, b
d, b
d, c
There are 12 permutations in all.
Evaluate $_4P_2$ to have:
$_4P_2= \dfrac{4!}{(4-2)!}
\\_4P_2=\dfrac{4!}{2!}
\\_4P_2=\dfrac{24}{2}
\\_4P_2=12$
