This document provides examples of counting problems and introduces fundamental counting principles like the multiplication principle and permutations. It explains how to calculate the number of possible outcomes for scenarios like making sandwiches, creating criminal identification profiles, forming school committees, generating license plates, and arranging letters and objects in different orders. Formulas for permutations with and without repetition are presented along with worked examples of applying these concepts.

2.  You
 You
want to make a sandwich:
have 4 types of meat (Ham,
Turkey, Roast Beef, Salami) and 3
types of bread (White, Wheat, Rye) to
choose from.
 How many different sandwiches can
you make?
(Draw a tree diagram!)

4.  Fundamental
Counting Principle
(FCP): Multiplying the number of
ways each event can occur gives
the number of possible outcomes.

5.  A criminal
identification kit contains
195 hairlines, 99 eyes, 89 noses,
105 mouths, and 74 chins.
 How many different faces can be
made?

6.  A high
school has 273 freshmen,
291 sophomores, 252 juniors, and
237 seniors.
 How many different ways can a
committee be formed that includes
1 person from each grade?

7.  A standard
New York
license plate has 3 letters
followed by 3 digits.
 If digits and letters can be repeated,
how many possibilities are there?

8. many if digits and letters can’t
be repeated?
 How

9.  How
many different 7 digit phone
numbers are possible if the first digit
cannot be 1 or 0?

10.  A multiple
choice test has 10
questions with 4 answer choices
each. How many different ways
could you complete the test?

11.  How
many different ways can you
arrange the letters A, B, and C?
 Make a list.
 Now,
use the FCP.

12.  The
# of permutations (orderings) of
n distinct objects is n!
 n! is read “n factorial”
 Factorial means:
n ∙ (n – 1) ∙ (n – 2) ∙ … ∙ 3 ∙ 2 ∙ 1
 Examples:
 5!
 8!

13.  In
how many different orders can
you complete 6 homework
assignments?

14.  Find
the number of distinct
permutations of the letters in each
word.
 IOWA
 FLORIDA

15.  How
many ways can you line up 9
people for a picture?

16.  The
number of permutations of r
objects from a group of n distinct
objects is denoted nPr.
nPr

17.  You
are considering 10 colleges. In
how many orders can you visit 6 of
them?
 All
10 of them?

18.  There
are 12 books on the summer
reading list. In how many orders
can you read 4 of them?
 All
12 of them?

19.  There
are 9 players on a baseball
team. In how many ways can you
choose the batting order for all 9
players?
 In
how many ways can you choose
a pitcher, catcher, and shortstop
from the 9?

20.  If
certain objects repeat, they are
not distinct anymore.
 To find these permutations with
repetition where n is the # of objects
and q is the number of times any
object repeats is:

21.  Find
the number of distinguishable
permutations of the letters in:
 OHIO
 MISSISSIPPI

22.  Your
dog has 8 puppies, 3 are male
and 5 are female. How many
different birth orders are possible?
(Hint: One is MMMFFFFF)

23.  A music
store wants to display 3
identical keyboards, 2 identical
trumpets, and 2 identical guitars.
How many distinguishable displays
are possible?