Lesson # 57: Permutations Involving Different Objects

Objectives: to introduce the concept of permutation
                  to find the number of permutations of n district objects taken r at a time

example: Two letter, A and B, can be written in two different orders, AB and BA. These are permutations of A and B.
a)  List all of the permutations of 3 letter A, B and C.
ABC, ACB, BAC, BCA, CAB, CBA
                 3 x 2 x 1 = 6
b) How many permutations are there for 4 letters A, B, C and D.
             4 x 3 x 2 x 1 = 24
c) Predict the number of permutations of 5 letters A, B, C, D and E
           5 x 4 x 3 x 2 x 1 = 120        5! (factorial)

example: Consider the letters A, B, C, D and E.  Instead of using all the letters to from permutations, we could use fewer letters.  For example, DB is a 2-letter perutations of these 5 letters.
a) List all the different 2-letter permutation of the 5 letter A, B, C, D and E.
AB AC AD AE     DA DB DC DE                        cP2
BA BC BD BE      EA EB EC ED    = 20              5P2 = 20
CA CB CD CE            

5 x 4 = 20    5P2 =    5!       = 5 x 4 x 3 x 2 x 1      EQUATION:  nPr =  n!
                               (5-2)!                  3 x 2 x 1                                       (n-r)!

example: How many different permutation can be formed using all of the letter of the word COMPUTE?
7P7  or   7! = 5040

Lesson # 56: The Fundamental Counting Principles

Investigate Counting without Counting:

example:    E        H       2
                 M   J   C       3
                  Y       P       2
2 x 3 x 2 = 12  (there are 12 possible meals)

The Fundamental Counting  Principle:
if one item can be selected in m ways, and for each way a second item can be selected in n ways, then the two items can be selected in m x n ways.

example part 1: how many different 2-digit numbers are there?
   9   x   10   = 90
1,2,3,4,5        0,1,2,3,4
6,7,8,9           5,6,7,8,9

example part 2: repetitions are not allowed.
   9   x   10   = 81
1,2,3,4,5        0.1.2.3.4
6,7,8,9           5,6,7,8,9

example: how many different patterns are possible for the answers to the 7 questions of the test (2 answers each)
2 x 2 x 2 x 2 x 2 x 2 x 2 = 1/128