Counting, Permutations,

1
Counting, Permutations, Combinations
2
A counting problem asks how many ways some
event can occur.

• Ex. 1 How many three-letter codes are there
  using letters A, B, C, and D if no letter can be
  repeated?
• One way to solve is to list all possibilities.

3
Ex. 2 An experimental psychologist uses a
sequence of two food rewards in an experiment
regarding animal behavior. These two rewards are
of three different varieties. How many different
sequences of rewards are there if each variety
can be used only once in each sequence?
Next slide
4

• Another way to solve is a factor tree where the
  number of end branches is your answer.

5
Fundamental Counting Principle
Suppose that a certain procedure P can be broken
into n successive ordered stages, S1, S2, . . .
Sn, and suppose that S1 can occur in r1
ways. S2 can occur in r2 ways. Sn can occur
in rn ways. Then the number of ways P can occur
is
6
Ex. 2 An experimental psychologist uses a
sequence of two food rewards in an experiment
regarding animal behavior. These two rewards are
of three different varieties. How many different
sequences of rewards are there if each variety
can be used only once in each sequence? Using the
fundamental counting principle
3
2
7
Permutations
An r-permutation of a set of n elements is an
ordered selection of r elements from the set of n
elements
! means factorial Ex. 3! 321
0! 1
8
Ex. 1How many three-letter codes are there using
letters A, B, C, and D if no letter can be
repeated? Note The order does matter
9
Combinations
The number of combinations of n elements taken r
at a time is
Order does NOT matter!
Where n r are nonnegative integers r lt n
10
Ex. 3 How many committees of three can be
selected from four people? Use A, B, C, and D
to represent the people Note Does the order
matter?
11
Ex. 4 How many ways can the 4 call letters of a
radio station be arranged if the first letter
must be W or K and no letters repeat?
12
Ex. 5 In how many ways can our class elect a
president, vice-president, and secretary if no
student can hold more than one office?
13
Ex. 6 How many five-card hands are possible from
a standard deck of cards?
14
Ex. 7 Given the digits 5, 3, 6, 7, 8, and 9, how
many 3-digit numbers can be made if the first
digit must be a prime number? (can digits be
repeated?)
Think of these numbers as if they were on tiles,
like Scrabble. After you use a tile, you cant
use it again.
15
Ex. 8 In how many ways can 9 horses place 1st,
2nd, or 3rd in a race?
16
Ex. 9 In how many ways can a team of 9 players
be selected if there are 3 pitchers, 2 catchers,
6 in-fielders, and 4 out-fielders?