The document discusses the fundamental principle of counting and provides examples of how to use it to calculate the number of possible outcomes in different scenarios. The principle states that if event A can occur in m ways and after A occurs event B can occur in n ways, then the total number of ways the two events can occur is m×n. Examples include calculating the number of ways to draw balls from a jar to make a sum, possible Facebook profile layouts using different applications, password combinations, and arrangements of friends that could appear in a Facebook profile box.

2. Fundamental Principle ofFundamental Principle of
CountingCounting
• Definition:Definition: Let A and B be two events. SayLet A and B be two events. Say
event A can occur inevent A can occur in mm different ways. After Adifferent ways. After A
has occurred, say event B can occur inhas occurred, say event B can occur in nn
different ways. Then the number of ways thatdifferent ways. Then the number of ways that
the two events can occur isthe two events can occur is mm··nn..

3. Basic Counting ProblemBasic Counting Problem
• Say that I have a jar with 7 balls in it numbered 1
through 7. How many ways can 10 be made (by
addition of the numbers on the balls) if I pull a ball
out of the jar, put it back, and then draw out a
second ball.

4. FUNDAMENTALFUNDAMENTAL
PRINCIPLE OFPRINCIPLE OF
COUNTINGCOUNTING

5. • EXAMPLE: (like #9,11 on HW)EXAMPLE: (like #9,11 on HW)
• You want to redesign your FacebookYou want to redesign your Facebook
profile and change the layout to seeprofile and change the layout to see
only 3 applications. You want oneonly 3 applications. You want one
application that pertains to yourapplication that pertains to your
friends, one that pertains to music,friends, one that pertains to music,
and one that pertains toand one that pertains to
pictures/videos.pictures/videos.
• If these are the applications you haveIf these are the applications you have
available in each category, how manyavailable in each category, how many
layouts are possible:layouts are possible:
– MusicMusic: 1. iLike: 1. iLike
2. My Band2. My Band
3. Profile Song3. Profile Song
– FriendsFriends: 1. Top Friends: 1. Top Friends
2. Compare Friends2. Compare Friends
– Pictures/VideosPictures/Videos: 1. My Flickr: 1. My Flickr
2. Slideshows2. Slideshows
3. Graffiti3. Graffiti
4. Animoto4. Animoto
VideosVideos
– Answer: 3x2x4=24Answer: 3x2x4=24

6. • EXAMPLE:EXAMPLE: (like #19 on HW)(like #19 on HW)
• Your little brother just broke into your Facebook account andYour little brother just broke into your Facebook account and
now you need to change your password. You want it to be 5now you need to change your password. You want it to be 5
characters long. The first 3 characters are to be letters andcharacters long. The first 3 characters are to be letters and
the last 2 characters are to be numbers. How many passwordsthe last 2 characters are to be numbers. How many passwords
are possible? What if you don’t want to repeat letters andare possible? What if you don’t want to repeat letters and
numbers?numbers?
• Answer: 26x26x26x10x10=1,757,600Answer: 26x26x26x10x10=1,757,600
• Answer: 26x25x24x10x9=1,404,000Answer: 26x25x24x10x9=1,404,000

7. II. Fundamental Principle of Counting
• EXAMPLE:
– Say that you have 42 friends
on Facebook. Any time you
access your profile, 6 of your
friends will appear in a box
below your picture. Each
time these friends are
randomly generated out of
your 42 friends. How many
different arrangements of 6
people can show up?
– Answer:
• 42x41x40x39x38x37=
3,776,965,920

8. II. Fundamental Principle of Counting
• EXAMPLE: (like #15,19 on HW)
– What if your friend Ally or
boyfriend Bobby must be in
position one, that is, they
are your only 2 choices for
that first spot?
(note: you still have 42 friends)
– Answer:
• 2x41x40x39x38x37=
179,855,520

9. Just to note…
• This can also be thought of as: nPr
• Permutations of n objects taken r at a time.
• For this example we would have 42P6 which
would get you the same answer

10. Just to note…
• This can also be thought of as: nPr
• Permutations of n objects taken r at a time.
• For this example we would have 42P6 which
would get you the same answer