Transcript 10th_Ch4_Lect42

Chapter 4
Lecture 4
Section: 4.7
Counting
Fundamental Rule of Counting:
If an event occurs m ways and if a different event occurs n
ways, then the events together occur a total of m·n ways.


Let us recall our example of rolling 2 dice. How many
possible outcomes are possible. We know that one die has
six sides and since we have two of them, then by the
fundamental rule of counting we get
6×6=36.
Also recall a husband and wife that want to have 3 children.
Since at each birth there are two possible outcomes (boy,
girl), then the number of different combinations of births is
2×2×2=8
Example #1:
You have 3 shirts, 5 pair of shoes and 6 pair of pants. How many
different outfits can be made from the given information?
3·5·6=90
2. An ATM code consists of only 4 digits. How many different
codes are possible?
10·10·10·10 = 104 = 10,000
3. At CSU-Long Beach, the password to log into www.my.csulb.edu
consists of 2 letters and then 4 digits. For example, ab1234 is a
password. How many different passwords are possible.
4. How many different combinations of heads and tails can be made
if you flip a coin 4 times?
5. What if we had a 5 digit home security code that had an additional
property that digits could not repeat. How many different codes are
possible?
6. A UPS man has 7 locations to make deliveries. How many different
routes are possible to make all of his deliveries?
In this case we would have to use the Factorial Rule. n!
Where n is the number of items that can be arranged.
n! n(n  1)( n  2)( n  3) 3  2 1
7. If we wanted to rank the top 10 movies of 2010, how many
possible outcomes are there?
Example 8: In a state lottery, a player wins or shares in the jackpot
by selecting the correct 6-number combination when 6 different
numbers from 1 through 42 are drawn. If a player selects one
particular 6-number combination, how many arrangements of 6
numbers out of 42 total numbers are possible.
In this case we will use the method called the Combination Rule.
We must have a total of n different items available. (42)
We must select r of the n items (6 of 42).
We must consider rearrangements of the same items to be the same.
That is the arrangement 123456 = 654321 = 512346 = 321654 and so on.
This case tells us that the order of the outcome does not matter.
Combination Rule Formula. ( Order is not taken into consideration)
n!
n Cr 
(n  r )! r!
So the answer to our question n=42 and r=6.
42!
 5245786
42 C6 
(42  6)!6!
Question: What is the probability of winning the lottery if to win you
pick 6 numbers out of 42.
1
1
7

 1.906 x 10
5245786
42 C6
9. What if the order of the numbers does matter? Better said, what
if the order of the numbers is taken into consideration? We saw in
the previous example that the six numbers 123456 was the same as
654321. However, if we take into consideration the order of the
numbers, then 123456 is not the same as 654321 because the way
the numbers are ordered are totally different.
Permutation Rule: (Order is taken into consideration)
n!
n Pr 
(n  r )!
So if we take into consideration the order that the numbers are
drawn, then
42!
 3776965920
42 P6 
(42  6)!
10. Say I have 4 markers in my book bag. I want to only select 2 of
the 4 markers.
Markers: B(black), R(red), G(green), P(purple)
4
C2  6
4
P2  12
B,R
B,R
R,B
B,G
B,G
G,B
B,P
B,P
P,B
R,G
R,G
G,R
R,P
R,P
P,R
G,P
G,P
P,G
11. Recall the example:
What if we had a 5 digit home security code that had an additional
property that digits could not repeat. How many different codes are
possible?
12. In how many ways can a sorority of 20 members select a
president, vice president and treasury, assuming that the same person
cannot hold more than one office.
13. In a horse race involving 10 horses, how many ways can first,
second, and third place be decided?
14. A certain department consists of 10 males and 8 females. How
many different ways can this department form a committee of
members consisting of:
a. 5 people.
b. 3 male and 2 female.
15. In a class of 40 students, how many ways can a study group of 6
students be arranged?