"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
