FUNDAMENTAL COUNTING, PERMUTATIONS, AND COMBINATIONS

The Fundamental Counting Principle​
If there are n items to choose from and m1 ways for the first choice of the items, m2 ways for the second choice, and so on, then there are a total of m1• m2• m3,…mn ways to choose n items.

Example 1
For breakfast there are 3 types of breakfast burritos, 2 types of chips and 5 types of drinks. How many breakfast meal choices comprising of the three components are there?

Solution
# of burritos  times  # of chips    times # of drinks  equals  # of choices
          3              x            2              x          5                 =                30

The Factorial
The factorial is the product of a number and all the positive integers less than it. The factorial symbol is denoted by!. Note that 0!=1

Example:
5!= 5x4x3x2x1=120. Note 0!=1

Practice Excercises
​\(\begin{array}{l}{\rm{Evaluate}}\\1.\,\,\,\,5!\\2.\,\,\,\,\frac{{4!}}{{0!}}\\3.\,\,\frac{{1000!}}{{999!3!}}\\4.\,\,\frac{{9!}}{{7!3!}}\end{array}\)

Permutation
A Permutation is the number of ways you can make selections (place or arrange) in such a way that order is important. Permutations can also be thought of as ordered grouping of elements of a set
Tip: Think of Permutation as placement or position!!!

Formula:\({}_n{P_r} = P\left( {n,r} \right) = \frac{{n!}}{{\left( {n - r} \right)!}}\)

Combination
A combination  is the number of ways you can make selections(choose) in such a way that order is not important.Combinations can also be thought of as un-ordered grouping of elements of a set
Tip: Think of combination as choice!!!
Formula: ​\({}_n{C_r} = C\left( {n,r} \right) = \frac{{n!}}{{\left( {n - r} \right)!r!}}\)

Circular Permutation
The number of ways n objects can be arranged in a circle is given by  (n-1) !

Alphabetical Permutations
The number of arrangements of  an n lettered word with r, s, and t number of letters repeated is given by: ​\(\frac{{n!}}{{r!s!t!}}\)

Practice Problems
​1.How many ways can 5 campers be arranged around a camp fire?
2.How many arrangements of the letters of the word Hermosillo are there?
3.For the letters in the list A,B,C,D
    a)List all the two letter combinations and confirm your result using the formula.
    b)List all the two letter arrangements and confirm your results using the formula.