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Fundamental Counting Principle
There are several techniques for counting the number of ways an event can occur. One is
the Fundamental Counting Principle which is used to find the number of ways 2 or more
events can occur in sequence.
The Fundamental Counting Principle
If one event can occur in m ways and a second event can occur in n ways, then the
number of ways the two events can occur in sequence is m*n.
(This rule can be extended for any number of events occurring in sequence.)
Examples:
1. You are purchasing a new car. Using the following manufacturers, car sizes, and colors,
how many ways can you select a car?
Manufacturer:
Ford, GM, Chrysler
Car size:
small, medium
Color:
white, red, black, green
2. You increase you choices above to include a Toyota, a large car, and a tan or gray car.
How many ways can you choose your car now?
3. The access code for a car’s security system consists of four digits. Each digit can be 0
through 9. How many access codes are possible if:
each digit can be repeated?
each digit can be used only once and not repeated?
4. How many license plates can you make if a license plate consists of:
six letters, each of which can be repeated?
six letters, each of which cannot be repeated?
two letters, followed by four numbers, with no restrictions?
two letters, followed by four numbers, where the letters O and I are not used
and the first number cannot be 0, with no other restrictions?
Permutations
An important application of the Fundamental Counting Principle is determining the number of
ways n objects can be arranged or a permutation (in order)
A permutation is an ordered arrangement of objects
The number of different permutations of n distinct objects is n! which means n factorial

calculated as follows:
3! = 3*2*1 = 6
4! = 4*3*2*1
! key is located: Math-PRB-#4
Examples of Finding the Number of Permutations of n Objects
1. The starting lineup for a baseball team consists of nine players. How many different
batting orders are possible using the starting lineup?
2. There are four teams in the NFC East, namely the Eagles, Cowboys, Redskins, and
Giants. How many different final standings are possible for this division of the NFL?
3. There are 8 students left in a kindergarten classroom. How many different ways can the
students get in line to go out to recess?
Suppose you want to choose just some of the objects in a group and put them in order. This
situation is called a Permutation of n Objects Taken r at a Time.
The number of permutations of n objects taken r at a time is:
n
n = total objects
Pr 
n!
 n  r !
r = how many needed
Examples:
4. In a race with eight horses, how many ways can three of the horses finish first, second,
and third place?
5. The board of directors for a company has twelve members. How many ways can they
assign the positions of president, vice-president, secretary, and treasurer?
6. Find the number of ways of forming a three-digit PIN code in which no digit is repeated.
Distinguishable Permutations
Sometimes you need to find how many possible ways you can order a set of objects where
some of the objects are the same. For example,
How many ways can you order the 7 letters AAAABBC?
There are 7! = 5040 ways to arrange 7 objects. BUT not all of these
permutations are different, or distinguishable.
We can find the number of distinguishable permutations by using the following formula:
The number of distinguishable permutations of n objects where n1 are
one type, n2 are another type, and so on is:
n!
n1 ! n2 ! ... nn !
So to find the number of distinguishable ways to order the 7 letters above:
7!
7 6 5 4 3 2 1 210


 105
4! 2! 1! 4 3 2 1 2 1 1
2
Examples
1. A building contractor is planning to develop a subdivision. The subdivision is to consist of
six one-story houses, four two-story houses, and two split-level houses. In how many
distinguishable ways can the houses be arranged?
2. The contractor wants to then plant six oak trees, nine maple trees, and five poplar trees
along the subdivision street. If the trees are spaced evenly apart, in how many
distinguishable ways can they be planted?
3. In how many distinguishable ways can the word Mississippi be written?
Combinations
Sometimes you want to choose some number of objects from a group where the order of the
objects does not matter. For example,
How many selections of 3 CD’s can you buy from a group of 5 possibilities?
Call the CD’s A, B, C, D, and E. What are the different sets of 3 you can choose?
ABC
ACD
BCD
CDE
ABD
ACE
BCE
ABE
ADE
BDE
How many choices are there? ________
Notice that the selection with ABC only appears once as any other order of ABC
would be the same selection, such as:
The number of ways to choose r objects from n objects where order does NOT matter is
called a combination.
The formula for combinations is:
(Math-PRB-3)
n
Cr 
n!
 n  r  !r !
Use the formula to find the number of CD selections from the above problem.
5
C3 
5! 5 4 3 2 1 20


 10
2!3! 2 1 3 2 1 2
Other Examples
1. PENNDOT plans to develop a new section of 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 that placed a bid?
2. The manager of an accounting department wants to form a three-person committee from
the 12 employees in the department. In how many ways can the manager do this?
3. Find the total number of possible 5-card poker hands that can be dealt from a standard
deck of cards.
Permutations & Combinations
Prob/Stat
Name__________________________
Date _______________ Period _____
Read each problem carefully to determine which type of problem it is. Your choices
are n!, nPr, nCr, and distinguishable permutations. You may use the TI-83 calculator to
get the answers—make sure you write what you typed into it.
1. How many three person committees can be chosen from a group of eight people?
2. In how many ways can a president and vice-president be selected in a class of 16
students?
3. How many ways can five students stand in a line for a photo?
4. How many four letter codes are possible if you have the letters A, B, C, D, E, and F to
choose from if they may not be used again once chosen?
5. There are 10 players on a basketball team. How many ways can a starting line-up of five
players be chosen?
6. There are 7 things in a hat. How many ways can you grab a handful of 5 things from the
hat at once?
7. If there are three seats available on the bus and two people want to sit down, how many
different ways can those two people arrange themselves in the empty seats?
8. How many different 3 letter access codes are possible using the last 5 letters of the
alphabet if repetition is forbidden?
9. Andrew has seven different kinds of cars to use on his model railroad, not including the
engine. In how many different orders can he arrange the cars behind the engine?
10. If you were given a choice of questions on a test, how many pairs of questions can you
choose if there are four questions to choose from?
11. A costume designer has 5 zombies, 7 ghouls, and 4 corpses to dress for a scene in a
new movie being filmed. In how many distinguishable ways can she get them ready for the
shot?
12. Proteins are made from linear sequences of amino acids. How many different proteins
could be made from the amino acids phenylalanine, glutamic acid, and lysine?
13. How many ways can you order the cards in one suit of a standard deck?
14. How many distinguishable ways can you arrange the letters in the word divided?
15. If there are 30 desks in a classroom, how many ways could you making a seating chart
for a class of 24 students?
Counting Principle Review
Prob/Stat
Name______________________________
Date _________________ Period _______
1. A certain lock combination consists of a sequence of 5 letters of the alphabet. How many
different codes are possible if each letter can be repeated?
If the letters cannot repeat?
If the combination is three letters and two numbers, and there are no repeated digits?
2. A paper boy has to deliver papers to 12 houses on a street. How many different ways can
he stop at the houses?
3. The Board of Health must inspect 15 restaurants this week. How many different ways can
an inspector select three of these restaurants to investigate on Monday if he will visit three
per day?
4. How many distinguishable permutations of the letters in the word CONFIDENCE are
there?
5. A day care worker must choose toys from a basket containing 10 toys to put out on the
floor for children to play with for the next hour. How many selections of 6 toys can she
make?
6. A European tour includes 4 stopovers to be selected from 10 cities.
How many tours are available if the order of the stopovers is taken into consideration
when choosing a tour?
How many tours are available if the choice you have is the 4 cities, but not the order
you will visit them?
7. The manager of an accounting department wants to form a five-person committee from
the 18 employees in the department. In how many ways can the manager do this?
8. How many ways can a TV station schedule a random selection of 8 commercials from a
set of 16 from a particular sponsor for an hour program?
How many different groups of 8 commercials do they have to choose from if they
are going to worry about what order to play them in later?
9. Forty-three race cars started the Daytona 500. How many ways can the cars finish
first, second, and third?
10. How many distinguishable ways can a florist line up 6 potted roses, 8 potted irises,
and 5 potted tiger lilies in the store window?