Transcript Combinations & Permutations

It is very important to check that we have
not overlooked any possible outcome.
One visual method of checking
this is making use of a
tree diagram.
Ex. Flip a coin, then roll a die
S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
The Counting Principle: The total number of
possibilities for two or more independent events
is the product of the number of possibilities for
each event.
Example:You want to create a computer code
using the letters A, B, C, D, E, and F. If letters
may be re-used how many possible codes are
there to choose from?
At Johnny’s Burger Place, a customer can get a
customized meal by ordering either a turkey burger,
chicken burger, hamburger, or garden burger with a side
order of potato chips or french fries with a choice of either
juice, milk, or soda.
•Use a tree diagram to list all the different combinations
of a burger, side order and a drink.
•Describe ways and give examples of how Johnny could
change his menu so that a customer would have 30
different choices.
Permutations changing the order of
elements arranged in a particular order.
(ORDER MATTERS!)
Example: Using the word BAT, how
many three letter combinations can be
made? (order matters and no letter may
be repeated)
10P3
= 10 • 9 • 8 = 720
Factorial (!)  the product of a given positive
integer multiplied by all lesser positive integers.
This is a case of permutations where all of the
objects are used.
Example:You want to create a computer code
using the letters A, B, C, D, E, and F. This time
letters may only be used once. How many
possible codes are there to choose from?
1) If you have a combination lock that contains only the numbers
from 0 to 9, and the combination contains three numbers, how
many possible combinations exist for this lock (assume numbers
can repeat)?
10 • 10 • 10 = 103 = 1000
2) There are 7 books on a shelf. How many different ways can you
arrange them?
7 • 6 • 5 • 4 • 3 • 2 • 1= 7! = 5040
3) How many different ways can we arrange the letters in the word
MATH?
4 • 3 • 2 • 1= 4! = 24
P
=
n k
n!
(n  k)!
n : total number of objects in a group
k : total number of objects taken
from n
Example: If six divers are entered
into a competition, how many
possibilities are there for the top
three places? (remember order
matters!)
If 40 names are placed in a hat,
how many permutations could be
made if 15 names are selected?
(assume order matters because of
the different prized awarded)
40!
40!
22

 5.26 10
40 P15 
40  15! 25!
Combinations  the arrangement of
elements into various groups without
regard to their order in the group.
Example: Using the word BAT, how many
two-letter combinations can be made?
(remember order doesn’t matter!)
C
=
n k
n!
k!(n  k)!
n : total number of objects in a group
k : total number of objects taken
from n

Example: With 32 seeds at Wimbledon (a
famous tennis tournament in Europe), how
many two player combinations are there for the
final match?
49!
49  48  47  46  45  44

 13,983,816
49 C6 
6!(49  6)!
6  5  4  3  2 1
1) How many different ways can you eliminate all of the 16 balls
from a pool table (assuming that hitting the 8 ball in early doesn’t
end the game like real pool)? Order matters!
16! = 2.09 •
13
10
2) How many ways can first and second place be awarded to 10
people?
10P2
= 10 • 9 = 90
3) Using the word numbers:
(a) If order matters, how many different arrangements are there for
all letters in numbers?
7! = 5040
(b) If b was definitely the first letter, now how many possible
arrangements are there?
6! = 720
4) You have 5 shirts, but you will select only 3 for your vacation. In
how many different combinations of shirts can you bring?
5!
5 43

 10
5 C3 
3!(5  3)! 3  2  1