Combinatorics is the study of counting techniques. In probability is helps to find the size of the sample space.

1. Know how to compute the n-th factorial.

The n-th factorial is the product of all positive integers (counting numbers, 1, 2, 3, 4, etc.) from 1 to the n-th integer.

Examples: 5! = 5 x 4 x 3 x 2 x 1 = 120

7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040

The n! = n x (n-1) x (n-2) x ...x 2 x 1

By definition 1! = 1 and 0! = 1

The Number of Ways to Sequence Items 

For some experiments you might consider the number of sequences in which items could be arranged.

Example: What is the sample space or possible sequences in which you can arrange the first 10 letters of the alphabet?

The principle of multiplication states that the number of possibilities or sequence of n items is: n!

So the number of possible ways to sequence the first 10 letters of the alphabet is 10! = 3,628,800.

Example: What is the Pr[ABCDEFGHI] in that sequence, since the event of getting the letters in that order is 1 and the total possiblities is  10!, Then Pr[ABCDEFGHI] = 1 / 10!
 

2. Know how permutation related to sequencing of objects.

The permutation is the possible arrangements of items according to the their sequence.

No item can appear more than once in a single sequence.

So ABC and CBA are different permutations that happens to involve the same three letters.

3. Know how to find the number of permutations of r items from a set of n different items.

The number of permutations are computed from the following formula:

, where r is the permutation size, n is number of items to select from.

Example if you need to select 4 (r) people from a set of 6 possible candidates for office where the order of selection is important, Jim, Sam, Eve is different from Eve, Sam, Jim.

,

360 is the permutation of sample space.

4. Know how to find the number of combinations of r items from a set of n different items.

The collection of r items taken from a set of n items where the particular items are included and not their sequence is called combination.

Example the following set of letters are all the same combination:

ABC , ACB, BAC, BCA, CAB, CBA.

The number of combinations of r objects taken from n objects,  , can be determined from the formula:

, where r is the combination size, n is number of items to select from.

Example if you need to select 4 (r) people from a set of 6 possible candidates for office where the order of selection is not important, Jim, Sam, Eve is the same as Eve, Sam, Jim.

,

15 is the combinations of sample space.