"To permute" means "to rearrange", so permutations
are ways to rearrange things.

The number of permutations of the 5 letters A,B,C,D,E taken
3 at a time is 60.  That's because there are 60 ways to pick
three letters from the 5 and arrange them in a certain order.
Here are all 60 permutations of the 5 letters A,B,C,D,E taken
3 ar a time.  Notice that there are 6 rows of 10 each:

ABC  ABD  ABE  ACB  ACD  ACE  ADB  ADC  ADE  AEB
AEC  AED  BAC  BAD  BAE  BCA  BCD  BCE  BDA  BDC
BDE  BEA  BEC  BED  CAB  CAD  CAE  CBA  CBD  CBE
CDA  CDB  CDE  CEA  CEB  CED  DAB  DAC  DAE  DBA
DBC  DBE  DCA  DCB  DCE  DEA  DEB  DEC  EAB  EAC
EAD  EBA  EBC  EBD  ECA  ECB  ECD  EDA  EDB  EDC

We could have calculated that there were 60 of them without listing 
them and counting them.  Here's how:

There are 5 ways to pick the letter to come first, A,B,C,D,or E.

For each of those 5 ways to pick the first letter, there remain 
only 4 unpicked letters that we could pick for the second letter.

That's 5×4 or 20 ways to pick the first two letters.

For each of those 5×4 or 20 ways to pick the first two letters, 
there remain only 3 unpicked letters that we could pick for the 
third letter.

That's 5×4×3 or 60 ways to pick all three letters.

So we write 5P3 = 60.  Sometimes we write P(5,3) = 60.

I hope this helps you understand what a permutation is, and how 
to calculate how many there are without listing them all.

Edwin