Labeling, Factorial, Combination, Permutation

Multiplication rule of counting
If there are n possible outcomes of event E1 and m possible outcomes for event E2, then there are a total of n × m possible outcomes for the series of events E1 followed by E2.

If there are more than two events, the first event has n1 possibilities and the second event has n2 and the third has n3, and so forth, the total number of possibilities of the sequence will be n1x n2 xn3…..nk

Labeling
Multinomial Formula: used for labeling problems in assigning k different labels to n members, with n1 labels of the first type, n2 labels of the second type, etc. ( note: n = n1 + n2 +…+ nk)

The total number of possibilities = n!/(n1!n2!…nk!)

Factorial Notation:

n-factorial = n! = n x (n-1) x (n-2) x (n-3) x.. x 2 x 1

Combination
Combination Formula is used for problems in choosing r objects from n total objects, where the order of the r objects listed does not matter

nCr= n! / [(n-r)! x r!]

Permutation
Permutation Formula isused for problems that r objects are selected from n objects and order of r objects does matter.

nPr = n! / (n-r)!

Steps to determine which approach to take:

1. Couting tools only apply if there is a finite number of outcomes
2. If assign n members to n slots, use n factorial
3. If count the number of ways to assign k labels to n member of a group, use multinomial formula
4. If count the number of ways to choose r objects from n objects and not concern about the order, use combination formula
5. If count the number of ways to choose r objects from n objects and concern about the order, use permutation formula
6. If counting tools cannot be used, count the possibilities one by one.