Answer
$P(5, 2)=20$
The list of all arrangements:
$ab,ac,ad,ae\\
ba,bc,bd,be\\
ca,cb,cd,ce\\
da,db,dc,de\\
ea,eb,ec,ed$
Work Step by Step
If we want to choose $r$ elements out of $n$ regarding the order, not allowing repetition, we can do this in $P(n, r)=\frac{n!}{(n-r)!}$ ways.
Hence,
$P(5, 2)=\dfrac{5!}{(5-2)!}=\dfrac{5!}{3!}=5\cdot4=20$
The list of all arrangements:
$ab,ac,ad,ae\\
ba,bc,bd,be\\
ca,cb,cd,ce\\
da,db,dc,de\\
ea,eb,ec,ed$
