Consider the following graph. $($For each answer, enter a mathematical expression.$)$

A graph with $3$ vertices and $5$ edges is shown.

Vertex a is connected to vertex b by edge e$1,$ by edge e$2,$ by edge e$3,$ and by edge e$4.$

Vertex b is connected to vertex a by edge e$1,$ by edge e$2,$ by edge e$3,$ and by edge e$4$ and to vertex c by edge e$5.$

Vertex c is connected to vertex b by edge e$5.$

$($a$)$

How many paths are there from a to c$?$

$($b$)$

How many trails are there from a to c$?$

$($c$)$

How many walks are there from a to c$?$