The length of the shortest trip from $A$ to $B$ along the edges of the cube shown is the length of 3 edges. How many different 3-edge trips are there from $A$ to $B$?

[asy]

size(4cm,4cm);

pair a1, b1, c1, d1;

a1=(1,1);
b1=(0,1);
c1=(1.6,1.4);
d1=(1,0);

pair e1, f1, g1, h1;

e1=(0,0);
f1=c1-(a1-d1);
g1=b1+(c1-a1);
h1=e1+(g1-b1);

draw(a1--d1--e1--b1--a1);
draw(b1--g1--c1--a1);
draw(c1--f1--d1);
draw(g1--h1--e1,dotted+1pt);
draw(h1--f1,dotted+1pt);

label("$A$",e1,SW);
label("$B$",c1,NE);

[/asy]
There are 3 choices for the first move starting from $A$. Having made the first move, then there are 2 choices for the second move. Then there is just 1 choice for the third move. Thus, there are $3\times2\times1$ or $\boxed{6}$ paths from $A$ to $B$.