Download 3.4 Counting Principles

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3.4 Counting Principles
LEQ: What is the difference between a permutation and a combination?
Procedure:
1. The Fundamental Counting Principle:
a. Definition 1: ______________________________________________: If
one event can occur in m ways, and a second event can occur in n ways,
the number of ways the two events can occur in sequence is m x n. This
rule can be extended for any number of events occurring in sequence.
b. Examples 1 & 2: Using the fundamental counting principle:
1. You are purchasing a new car. Use the following manufacturers, car
sizes, and colors, how many different ways can you select one
manufacturer, one car size, and one color?
Manufacturer: Ford, GM, Chrysler
Car Size: small, medium
Color: white, red, black, green
2. How many license plates can you make if a license plate consists of:
a. Six letters each of which can be repeated?
b. Six letter each of which cannot be repeated?
2. Permutations:
a. Definition 2: A ______________________ is an ordered arrangement of
objects. The number of different permutations of n distinct objects is n!.
b. Example 3: Finding the number of permutations of n objects:
3. The starting lineup for a baseball team consists of nine players. How
many different batting orders are possible using the starting lineup?
c. Definition 3: The number of permutations of n distinct objects taken r at a
time is
d. Examples 4 & 5: Finding n pr :
4. Find the number of ways of forming three-digit codes in which no digit
is repeated?
5. Forty-three race cars started the 2004 Daytona 500. How many ways
can cars finish first, second, and third?
e. Definition 4: The number of ____________________________________
of n objects, where n1 are of one type, n2 are of another type, and so on is
f. Example 6: Distinguishable permutations:
6. A building contractor is planning to develop a subdivision. The
subdivision is to consist of 6 one-story houses, 4 two-story houses,
and 2 slit-level houses. In how many distinguishable ways can the
houses be arranged?
3. Combinations:
a. Definition 5: A _______________________ is a selection of r objects
from a group of n objects without regard to order and is denoted by n C r . The
number of combinations of r objects selected from a group of n objects
b. Example 7: Finding the number of combinations:
7. A state’s department of transportation plans to develop a new section
of interstate highway and receives 16 bids for the project. The state
plans to hire four of the bidding companies. How many different
combinations of four companies can be selected from the 16 bidding
companies?
3. Applications of Counting Principles:
a. Example 8: Finding probabilities:
A word consists of one M, four Is, four Ss, and two Ps. If the letters are
randomly arranged in order, what is the probability that the arrangement
spells the word Mississippi?
b. Example 9: Finding probabilities:
Find the probability of being dealt five diamonds from a standard deck of
playing cards. (In poker, this is a diamond flush).
4. HW: p. 157 (2, 3 – 39 mo3)