May 18, 2015
9.1
Basic Combinatorics
Objective:
To learn about countable sets (Multiplication Counting Principle,
Permutations, and Combinations)
Why: These methods allow one to count large sets of numbers
quickly.
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
How many different ways can three objects be arranged in
order?
A
B
C
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
A car manufacturing company asked 1000 people to view
their latest two-door coupe available in four colors (White,
Black, Red, and Silver). The viewers were asked to list their
color preference in order from favorite. How many ways
could the colors be ordered? Of the 1000 viewers how many
would you expect to choose the preference order White,
Black, Red, and Silver?
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
Multiplication Principle of Counting:
If a procedure P has a sequence of stages S1, S2, ... Sn and if
S1 can occur in r1 ways,
S2 can occur in r2 ways,
.
.
.
Sn can occur in rn ways,
then the number of ways that the procedure, P, can occur is the
product
r1r2...rn
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
Determine the possible number of license plates with
7 blanks for each condition.
1. The first three blanks are letters that can be
repeated and the last three are digits than can be
repeated.
2. How many different phone numbers are possible in
the 419 area code?
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
Permutation: number of ways to order a set of n objects
n! permutations of an n-set
Ex. How many ways can we line-up the senior class?
Ex. How many ways can we line-up four people from the senior
class?
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
Permutations of n objects taken r at a time:
P =
n r
n!
(n-r)!
Evaluate.
1. 8P6
2. 10P9
3. nP6
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
Count the number of different 7-letter "words" (don't
worry about whether they are in the dictionary) that
can be formed using the letters in each word.
F
C
L
A
O
R
L
I
L
D
A
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
Distinguishable Permutations:
If an n-set contains n1 objects of a first kind, n2 objects of
a second kind, and so on, with n1 + n2 + ... nk = n, then the
number of distinguishable permutations of the n-set is :
n!
n1!n2!n3!...nk!
M
I
S
S
I
S
S
I
P
P
I
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
Find the number of ways one can choose two marbles from
a bag containing three marbles:
1. if the order is important
2. if the order is not important
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
Combination Counting Formula:
The number of combinations of n objects taken r at a
time is denoted:
nCr =
n!
r!(n-r)!
Proof: Every permutation can be thought of as an unordered
selection of r objects followed by a particular ordering of
the objects selected.
nPr = nCr r!
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
Every NBA basketball team has 12 players. How many different
starting 5's are possible?
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
In the game of bridge, each player is dealt a 13-card hand
from a standard deck of 52 cards. How many different
card hands are possible?
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
A lottery requires winners to pick 6 integers between 1 and 45.
How many different lottery tickets are possible?
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
A pizzeria offers any combination of up to 8
different toppings. How many different pizzas
can be ordered if:
a. we choose any 3 toppings?
b. we choose any number of toppings (0 - 8)?
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
Formula for Counting Subsets of an n-set:
There are 2n subsets of a set with n objects (including
the empty set and the entire set).
May 18, 2015
Obj: To learn about countable sets
(Multiplication Counting Principle, Permutations, and Combinations)
HW:
(REG) (9.1) Pg.641: 1-41odd