(w/o Repetition) You are given a set with distinct objects.
See: Rosen, Section XX
Brualdi, Sections XX a (full) Permutation : an ordering of all of the objects
There are distinct arrangements of this type. a -Permutation : an arrangement using of the objects
There are distinct arrangements of this type. a -Combination : a selection of of the objects (order doesn’t matter)
There are distinct combinations of this type.

(with Repetition)
With Unlimited Repetition:
You are given a set with distinct objects. Each object may appear any # of times.
See: Rosen, Section 4.6
Brualdi, Sections 3.4 and 3.5 a -Permutation : an arrangement of length
The answer follows directly from the multiplication principle.
There are distinct arrangements of this type. a -Combination : a selection of of the objects (order doesn’t matter)
This is our “stars and bars” picture.
Think of a cash-drawer that uses
(See the next page.)
bars to create bins (for options)
There are
With Specified Repetition: distinct combinations of this type.
The object appears times. The total number of objects is given by a (full) Permutation : an ordering of all of the objects
If the objects were distinct, there would be distinct arrangements.
However, we must use the “division principle” to account for the fact that they aren’t distinct.
In the end, there are

distinct arrangements of this type.
Combinations with Repetition
Our “stars and bars” presentation is counting the number of possibilities for making selections from options.
For the example of , one possibility would be:
* * * I * * * * * * * I * * | | * * * * | * | * * * * *
objects in the first bin
objects in the second bin
objects in the third bin
objects in the fourth bin
objects in the fifth bin
objects in the sixth bin
objects in the seventh bin
Where we need a total of selections.
The number of distinct ways of selecting 22 objects from 7 possibilities is: