II) Combinatorial analysis
1°) Fundamental principle of combinatorial analysis:
If any procedure can be represented in different ways, if after that procedure a second
procedure can be represented in different ways and so on ... So the number of different
ways to perform this procedure in the given order is:
Example:
A license plate contains two distinct Latin letters followed by two numbers, the first of
which is non-zero. How many different plates can I print?
Answer:
For the first Latin letter there are 26 different ways to print it
For the second Latin letter we have 25 different ways to print it
For the first digit we have 9 different ways to print it
For the second digit we have 10 different ways to print it
For the third digit we have 10 different ways to print it
According to the fundamental principle of combinatorial analysis, the number of
plates to be printed is: plates.
2°) Permutation:
Let E be a set of n elements, a permutation on E is an order on E
Theorem:
There are n! permutations on E
Demo:
To swap the first element we have n possibilities
To swap the second element we have (n – 1) possibilities
To swap the third element we have (n – 2) possibilities
….
The total number of possibilities is
3°) Arrangement:
An arrangement of k elements among n other elements is an order on k «elements
chosen from these n elements noted by