Transcript Section 6-7 Permutations and Combinations File

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Algebra 2
Algebra 2
Permutations and Combinations
Lesson 6-7
Algebra 2
"My fruit salad is a combination of apples, grapes and bananas"
We don't care what order the fruits are in, they could also be
"bananas, grapes and apples" or "grapes, apples and bananas", its the
same fruit salad.
"The combination to the safe was 472". Now we do care about the
order. "724" would not work, nor would "247".
It has to be exactly 4-7-2.
Permutations are for lists (order matters) and
Combinations are for groups (order doesn’t matter).
Permutations and Combinations
Lesson 6-7
Additional Examples
Algebra 2
Permutations and Combinations
Lesson 6-7
Algebra 2
There are basically two types of permutation:
Repetition is Allowed: such as the “lock” example above. The
sequence of numbers could be "333".
No Repetition: for example a track race. You can't be first and
second at the same time!
Permutations and Combinations
Lesson 6-7
Algebra 2
Permutations and Combinations
Lesson 6-7
Algebra 2
Permutations and Combinations
Lesson 6-7
Additional Examples
Algebra 2
In how many ways can 6 people line up from left to right
for a group photo?
Since everybody will be in the picture, you are using all the items from the
original set. You can use the Multiplication Counting Principle or
factorial notation.
There are six ways to select the first person in line, five ways to select the
next person, and so on…
The total number of permutations is 6 • 5 • 4 • 3 • 2 • 1 = 6!.
6! = 720
The 6 people can line up in 720 different orders.
Permutations and Combinations
Lesson 6-7
Additional Examples
Algebra 2
Permutations and Combinations
Lesson 6-7
Additional Examples
Algebra 2
Permutations and Combinations
Lesson 6-7
Algebra 2
Algebra 2
Algebra 2
r
r
r
Algebra 2
Permutations and Combinations
Lesson 6-7
Algebra 2
Additional Examples
How many 4-letter codes can be made if no letter can be used
twice?
Method 1: Use the Multiplication Counting Principle.
26 • 25 • 24 • 23 = 358,800
Method 2: Use the permutation formula. Since there are 26 letters
arranged 4 at a time, n = 26 and r = 4.
26P4 =
26!
= 26! = 358,800
(26 – 4)!
22!
There are 358,800 possible arrangements of 4-letter codes with no duplicates.
Algebra 2
Permutations and Combinations
Lesson 6-7
Additional Examples
Algebra 2
Example: Imagine the yacht race, but with 5 yachts where the first 3
boats to cross the finish are deemed the winners. We wish to see how
many different outcomes will there be. In this case, the order in which
the three yachts cross the finish line DOES NOT matter. A selection in
which ORDER DOES NOT MATTER is a COMBINATION.
Permutations and Combinations
Lesson 6-7
Additional Examples
Evaluate 10C4.
10!
10C4 = 4!(10 – 4)!
10!
= 4! • 6!
10 • 9 • 8 • 7 • 6 5 4 3 2 1
= 4•3•2•1• • • • • •
6• 5• 4• 3•2•1
10 • 9 • 8 • 7
= 4•3•2•1
= 210
Algebra 2
Permutations and Combinations
Lesson 6-7
Algebra 2
Additional Examples
10C5
=
8C2
=
25C7
=
Permutations and Combinations
Lesson 6-7
Algebra 2
Additional Examples
A disk jockey wants to select 5 songs from a new CD that
contains 12 songs. How many 5-song selections are possible?
Relate: 12 songs chosen 5 songs at a time
Define: Let n = total number of songs.
Let r = number of songs chosen at a time.
Write: n C r = 12 C 5
Use the nCr feature of your calculator.
You can choose five songs in 792 ways.
Permutations and Combinations
Lesson 6-7
Algebra 2
Additional Examples
A pizza menu allows you to select 4 toppings at no extra
charge from a list of 9 possible toppings. In how many ways can you
select 4 or fewer toppings?
You may choose
4 toppings, 3 toppings, 2 toppings, 1 toppings, or none.
9C4
9C3
9C2
9C1
The total number of ways to pick the toppings is
126 + 84 + 36 + 9 + 1 = 256.
There are 256 ways to order your pizza.
9C0
Algebra 2