SOLUTION: how many different arrangements are there of the letters p,m,s,a

You can choose the first letter any of 4 ways.

For each of the 4 ways you can choose the first letter, you can choose the
second letter as any of 3 letters that are left.  

That's 4×3 ways you can choose the first 2 letters.

For each of the 4×3 ways you can choose the first 2 letters, you can choose the
3rd letter as either of the 2 letters that are left.

That's 4×3×2 ways you can choose the first 3 letters.

For each of the 4×3×2 ways you can choose the first 3 letters, you can only
choose the 4th letter as the only letter that is left.

That's 4×3×2×1 ways to choose the 4 letters.

4×3×2×1 = 24. That's also called "4 factorial", written 4!

Here they all are:

 1.  p,m,s,a
 2.  p,m,a,s
 3.  p,s,m,a
 4.  p,s,a,m
 5.  p,a,m,s
 6.  p,a,s,m
 7.  m,p,s,a
 8.  m,p,a,s
 9.  m,s,p,a
10.  m,s,a,p
11.  m,a,p,s
12.  m,a,s,p
13.  s,p,m,a
14.  s,p,a,m
15.  s,m,p,a
16.  s,m,a,p
17.  s,a,p,m
18.  s,a,m,p
19.  a,p,m,s
20.  a,p,s,m
21.  a,m,p,s
22.  a,m,s,p
23.  a,s,p,m
24.  a,s,m,p

Edwin