Combinatorics - Combinatorial Analysis

Variations or permuted combinations (permutations without repetition)
Combinations
Given a set of n different elements or objects. Select a subset of r elements out of n. Such selection is
called the combination.
A combination is an unordered arrangement of r objects selected from n different objects taken r at a time.
The number of distinct combinations selecting r elements out of n is
Therefore, combinations must differ from each other at least in one element.
Example:  Find the number of combinations of size 3 that can be made from digits 1, 2, 3, 4, 5 and write
them out.
Solution:  Since, n = 5 and r = 3 then
The combinations are,    1 2 3         2 3 4         3 4 5.
1 2 4         2 3 5
1 2 5         2 4 5
1 3 4
1 3 5
1 4 5
Example:  Find the number of combinations of size 4 that can be made from letters A, B, C, D, E, F and
write them out.
Solution:  Since, n = 6 and r = 4 then
The combinations are,    A B C D         B C D E         C D E F.
A B C E         B C D F
A B C F         B C E F
A B D E         B D E F
A B D F
A B E F
A C D E
A C D F
A C E F
A D E F
Combinations with repetition
The number of ways to choose r objects from a set of n different objects, so that an object can be chosen
more than once
Remember that combinations must differ from each other at least in one element.
Example:  Find the number of combinations of size 3 that can be made from digits 1, 2, 3, 4 if repetition is
allowed, and write them out.
Solution:  Since, n = 4 and r = 3 then
The combinations are,     1 1 1          2 2 2          3 3 3          4 4 4          1 2 3
1 1 2          2 2 1          3 3 1          4 4 1          1 2 4
1 1 3          2 2 3          3 3 2          4 4 2          1 3 4
1 1 4          2 2 4          3 3 4          4 4 3          2 3 4

Example:  Find the number of combinations of size 4 that can be made from letters A, B, C if repetition is
allowed, and write them out.
Solution:  Since, n = 3 and r = 4 then
The combinations with repetition are,     A A A A          B B B B          C C C C          A A B C
A A A B          B B B C          C C C A          B B A C
A A B B          B B C C          C C A A          C C A B.
A B B B          B C C C          C A A A
Variations or permuted combinations (permutations without repetition)
The variations of size r chosen from a set of n different objects are the permutations of combinations of r.
The number of variations of size r chosen from n objects equals the number of combinations of size r
multiplied by the r! permutations,
Example:  Find the number of variations of size 3 that can be made from digits 1, 2, 3, 4 and write them out.
Solution:  Since, n = 4 and r = 3 then
Notice that there are 4 combinations of size 3 chosen from the given 4 digits, each of which gives six
permutations as is shown below.
The variations are,          1 2 3           1 2 4           1 3 4           2 3 4
1 3 2           1 4 2           1 4 3           2 4 3
2 1 3           2 1 4           3 1 4           3 2 4
2 3 1           2 4 1           3 4 1           3 4 2
3 1 2           4 1 2           4 1 3           4 2 3
3 2 1           4 2 1           4 3 1           4 3 2.