Greetings 1st block students
Tonight's blog assignment involves everyone in first block.
All of you should click on comment and write a summary of today's lesson on permutations and combinations. Don't forget to include the lesson's objectives and some examples in your summary.
Happy Blogging.
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Today's lesson which is lesson 6.7 Combinatorics basically tells about the arrangement of items or things in which order does or does not matter. To begin lets learn some vocabulary. What is a permutation? A permutation is an arrangement of items in particular order. What is a combination? A combination is a arrangement of items in no particular order. It is also important to know about factorials and the multiplication counting principle.A factorial is denoted by an positive integer followed by an exclamation point. It tells you to mulitply the integer by every positive integer than the given integer.An example would be: 3!= 3 x 2 x 1 = 6. The multiplication counting principle uses the formula n!...You can use this principle instead of the permutation formula if you plan to choose all of the items in a particular set. Some permutations do not use all of the items availible in a set. You can still use the Multiplication counting principle or you may choose to use the permutation formula. An example of using the mult. counting formula would be a problem such as: Choose 3 officers from 26 people.You'd begin by multiplying by the beginning number 26 then multiply by 25,24 and so on.Formulas that are in important for this chapter are the permutation formula which is nPr= n!/(n-r)! and the combination formula which is nCr= n!/r!(n-r)!.When using permutations both items seleted and the order of their selection are important. When using combinations the only important thing is which items are selected. A problem you can use for an example of an permutation would be 1)In how many orders can 10 floats line up for the homecoming parade?
step 1)You will use 10! or to simplify without an calculator you will multiply (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1.) Your answer should result as 3628800. An example of an combination problem would be 12C3. You would use your formula and write 12!/3!(12-3)! Once you simplify you answer it should be 220. Conclusion: Always remember that when doing these problems if you approach the word "and" while using permutations it means to "multiply." If you approach the word "or" in an combination problem it means to "add" the solutions to the different combinations.
6.7
In today's lesson we learned about combinations and permutations. Through this we learned how to tell what was concidered a perutation(a batting line up) and what was considered a combination (a committee).
Vocab.
1)permutation-the arrangment of items in a preticular order
2)combination-an arrangment of items in no preticular order
3)factorial-denoted by positive intergers followed by !
Formulas:
the formula for permutation is
nPr=n!/(n-r)!
Ex. choose 3 officers from 26 people
26P3=26!/3!23!
26(25)24(23)/3(2)23! the 23s cancelso does the 24 with the 2 and 3
26(25)24=2600
The combination formula is:
n!/r!(n-r)!
Ex. 9C3
9!/3!(9-3)!
9!/3!(6)!
9(8)7(6)!/3!(6)!the 6s cancel so does the 9 with the 3 and the 8 with the 4
12(7)=84
Lesson 6-7: Combinatorics
Objectives: To learn about permutations and combinations. And to learn about n factorials.
What is a permutation? A permutation is an arrangement of items in a particular order. The numbers can be found by the mulitplication counting prinicple or the permutation formula.
What is a combination?
A combination is an arrangement of items where the order does not matter. Example: committees
What is a factorial?
A factorial is denoted by an intger followed by "!". Multiply integer by every positive integer smaller than the given number.
What is the multiplication counting principle?
It uses the n factorial formula. It is easy to use if choosing all items in the set.
Examples of factorials:
1. 3!= 3*2*1=6
2. 4! 4*3*2*1=24
Example of Multiplication Counting Principle (not using all items):
26*25*24
Example of Multiplication Counting Principle (using all items):
26!
Permutation Formula:
nPr=n/(n-r)
Example: n=26 r=3
26P3= 26!/(26-3)!
26P3=26*25*24*23!/23!
Your 23!'s cancel so you're left with 26*25*24!= 15,600
Use the permutation formula when you are not selecting all of the items available in a set. "Menu problems" contain the connecting word "and". "And" tells us to multiply the permutations.
Combination Formula:
nCr=n!/r!(n-r)!
Example: n=26 r=3
26C3=26!/3!(26-3)!
26C3=26*25*24*23!/3*2*23!
Your 23!'s cancel and your 6 and 24 reduce so you are left with
26*25*4=2600
In combination problems, the word "or" tells us to ADD the solutions to the different combinations.
The lesson today was 6.7 - Permutations and Combinatorics.
a Permutation is an arrangement of items in a very particular order. In most equations you can use factorials or MCP(multiplication counting principle) a factorial is a abreviated by the exclmation point such as n! . if n = 0 though, the factorial of n! = 1 kinda of like an exponent to the zero power. A combination is different than a permutation; combination's order does not matter.
For example 4! = 4x3x2x1 = 24
The Formula for Permutations is
nPr =
n!/
(n-r)!
The formula for Combinations is somewhat different there is an additional r factorial...
n!/
r!(n-r)!
For both equations n = the number of total objects and
r = the number you are actually selecting
Real world examples
For Permutations the line up of the boys tennis team where order matters.
For Combinations...
the random selection of songs on the iPod when you create a shuffle playlist.
Example of finding a permutation..
So 5 cars have to be washed at a local car wash since all of them are to be washed you can use factorial notation to find the answer 5! = 120 or you could use the Multiplication counting principle 5x4x3x2x1 = 120 . either way the answer is the same
Example of Combination..
Mrs Keeton requires 3 summer reading books. she gives a list of 20 different novels to read.
Jon wants to know how many different possibilities there is so he uses the formula and plugs in the required information.
20!/
3!(20-3)! = 1140 different combinations of books. Wow thats a lot.
this formula can be shortened and put into a TI-84 by using the fomula nCr under the math button.
Well thats basically all the information covered in this lesson short and sweet.
In lesson 6.7 the objective is to learn the formulas used to solve permutations and combinations. The vocabulary words for this lesson are permutation, combination, factorial, and the multiplication counting principle. A permutation is an arrangement of items in a particular order. A combination is an arrangement of items in no particular order. A factorial is denoted by a positive integer followed by an exclamation point, which tells you to multiply the integer by every positive integer that is smaller than the given integer. An example is: 4!= 4 x 3 x 2 x 1 = 24. The multiplication counting principle uses the formula n! You can use this principle instead of the permutation formula if you plan to choose all of the items in a particular set. Some permutations do not use all the items available in a set. You can still use the Multiplication counting principle or permutation formula. An example would be: choose 3 officers from 26 people. You would multiply 26 x 25 x 24 = 15600. The permutation formula is nPr= n!/(n-r)!. When using a permutation formula both items are selected and the order of their selection are important. An example would be: in how many orders can 10 floats line up for the homecoming parade? 10!= 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1. Which equals 3,628,800. The combination formula is nCr= n!/r!(n-r)!. When using a combination formula the only important thing is which items are selected. An example would be 26C3. Then you would solve 26!/3!(26-3)!
26!/(3!(26-3)!)
26!/(3!23!)
(26 x 25 x 24 x 23!)/((3 x 2)23!)
= 2600
Finally, in word problems, (or) means add; also, (and) means multiply.
COMBINATIONS AND PERMUTATIONS
In today's lesson we learned about how combinations are a mixture of items in any order, permutations are a mixture of items in a particular order, what an exclamation point has to do with math!!
Let's learn som....FORMULAS!
The Multiplication Counting Principle.
-multiplying by each number smaller than itself.
ex. 4!= 4 X 3 X 2 X 1= 24
-Permutation formula
nPr= n!/(n-r)!
ex. 3P2= 3!/(3-2)!
3 X 2 X 1/ 1!
=6
-Combination formula
(this formula is the same as the permutation formula EXCEPT you put an extra r! in the demonator.)
nCr= n!/r!(n-r)!
ex. 3C2= 3!/2!(3-2)!
= 3 x 2/ 2! 1!
= 3
Each of these formulas can be used for a real life example. You simply insert the numbers from word problems into the formulas. n will equal the total number of items and r will equal the number you are actually choosing. When there are multiple choices in the problem the word "and" means to multiply and the word "or" means to add.
*HINT*
*n will NEVER be smaller than r*
Simple enough? HAVE FUN!! =)
LESSON 6.7
this lesson deals with permutations and combinations. A permutation is an arrangement of items in a PARTICULAR order. A combination is an arrangement of items in NO PARTICULAR order. Use n Factorial to find your answers. for any positive integer n, n!=n(n-1)times...3 times 2 times 1. number of permutations of n items of a set arranged r items at a time is nPr.
nPr=n!/(n-r) for 0 is greater than or equal to r which is greater than or equal to n.
NUMBER OF COMBINATIONS
the number of combinations of n items of a set chosen r items at a time is nCr
nCr=n!/r!(n-r!)
EXAMPLE:
5C3=5!/3!(5-3)!=
5!/3!x2!=
120/6x2=
10
n=total # of objects
r=the # you're actually selecting.
0!=1
and=multiply
or=add
EXAMPLE:
choose 3 officers from 26 people
26x25x24
this problem can also be done using the permutations formula.
-ßLIИDVIδIOŊ
Lesson 6.7
Permutations and Combinations
What is a permutation?
-Permutation is an arrangement of items in a particular order. Find the numbers by the Mulitplication counting prinicple or the Permutation formula.
What is a combination?
-Combination is an arrangement of items where order does not matter. Ex.Committees and Athletic Teams where there is no differentation in skill or position.
What is a factorial?
-A factorial is denoted by a positive intger followed by an exclamation point it tells us to multiply the integer by every positive integer smaller than the given interger.
What is the multiplication counting principle?
-Uses the formula n! (factorial)if you plan to choose all of the items in a particular set.
*Some permutations do not use all the items availble in a set.*
FORMULAS:
Permutation formula
nPr= n!/(n-r)!
Combination formula
nCr= n!/r!(n-r)!
*In permutations both items selected and the order of their selection are important!
In Combinations the only important thing is which items are selected!*
EXAMPLES:
Pemutation
26P3= 26!/(26-3)!
26x25x24x23!/23!
Cancel the 23! Now you are left with 26x25x24=15,600
Combination
26C3=26!/3!(26-3)!
26x25x24x23!/3x2x23!
Cancel the 23! and multiply 3x2 then cancel the answer (6) from 24.
Left with 26x25x4=2600
Remember!
*The word andwhen using permutations means to multiply *
*The word orin a combination means to addthe solutions to different combinations*
*n will never be less than r*
Lesson 6-7
Permutations and Combinations
Lesson Objectives
-Use permutation to find specific orders of arrangements of objects.
-Use combinations to find the possible sequences of selections made in no particular order.
VOCAB:
Permutation- an arrangement of items in a particular order.
Combination- a selection in which order does not matter.
Factorials
Permutations and Combinations use factorials to find the amount of arrangements or selections that can be gained from a given amount of objects or situations.
A factorial is represented by an exclamation mark following an integer that must be positive. When a factorial is given it is understood that the integer shown is multiplied by every number below its value. Below is an example.
Ex: 3!=3*2*1
note: The factorial of 0 is 1.
Permutation.
Formula:
nPr= n!/(n-r)!
n = number of items
r = arranged set
zero must be less than of equal to r which must in turn be less than or equal to n.
Ex: Seven runners enter a race. First, second, and third places will be given to the three fastest runners. How many arrangements of first, second, and third places are possible with seven runners?
There are two methods to answer this question.
Method 1 is by using the Multiplication Counting Principle. This principle is the n! limited to r places.
In this case 7*6*5=210
Method 2
Use the permutation formula
7P3 = 7!/(7-3)!=7!/4!=210
There are 210 possible arrangements of first, second, and third places.
combinations
The formula:
nCr= n!/r!(n-r)!
n = number of items
r = set of chosen items
zero must be less than of equal to r which must in turn be less than or equal to n.
Ex: 12C3
12C3=13!/3!(12-3)!
=12!/3!*9!
=12*11*10*9!/3!*9!
=12*11*10/3*2
=220
There is another example where you must choose something such as a depleting selection and find how many ways it could be changed up. In this situation you would go down in order changing the value of the r as needed. Each would be solved for the combination and then added together to get the final answer.
-Justin Bennett
Todays lesson was focused on Permutations and Combinations:
Objectives:
To learn about permutations and combinations and to finf real world examples of each.
Vocabulary:
1) Permutation- is an arrangement of items in a paticular order.
2) Combination- is an arrangement of items were the order in which they are arranged is in no paticular order.
3) Factorial- denoted by positive intergers followed by !
formulas:
Factorial positive formula- n!=n(n-1)
Ex: In How many orders can 10 floats line up for the homecoming parade. 10!= 3,628,800
Permutation formula- nPr= n!/(n-r)!
Ex: 10P4= 10!/(10-4)!= 10!/6!= 5040
Combination formula- nCr= n!/r!(n-r)!
Ex: 5C3= 5!/3!(5-3)!= 51/3!*2!= 120/6*2=
Lesson 6.7 concerns Combinations and Permutation.
What is a Combination?
It is an arrangement of numbers or items in no particular order.
A combination's formula is written like so.
n!/r!(n-r)!
What is a Permutation?
A permutation in a methodically planned out series of numbers or items in a set
A permutation's formula is written like so:
nPr=n/(n-r)
What is a Factorial?
it is a function that instructs you to multiply by the next non negative integer in a series.
Example: 3!3x2x1=6
Lesson 6.7
In Lesson 6.7 we learned about permutations and combinations. We also learned about the formulas that go along with these.
A permutation is an arrangement of items in a particular order. You can find the number of permutations of of a set by using the Multiplication Counting Principle, or by using the permutation formula.
A comination is an arrangement of items where the order which they are arranged does not matter.Ex. committees and t-ball teams where there is no differnce in skill or position.
A factorial is denoted by a *positive* integer followed by an exclamation point. It tells you to multiply by the integer by every positive integer smaller than the given integer.
Example: 3!= 3x2x1=6
The Multiplication Counting Principle uses n!. You can use this principle instead of permutations formula. If you plan to choose all of the items in a particular set.
Example: Choose 3 officers from 26 people.
26x25x24
Formulas:
Permutations: nPr
n! divided by (n-r)!
Combinations: nCr
n! divided by r!(n-r)!
where n= total number of objects
r= the number you are actually selecting
Ex. How many ways can yoe choose a committee of 3 from 26?
26!/3!(26-3)!
26!/3!23!
which equals 26x25x24
-Lauren Thornton
In lesson 6.7, we learned about permutations, combinations, and what factorials are.
A permutation is an arrangement of numbers in a particular order. A combination is an arrangement of numbers in which the order of the numbers do not matter. You can denote a factiorial by using the exclamation point (!). In a factorial, you multiply the positive integer by every smaller positive integer.
An example of a factorial is-
5!=(5)(4)(3)(2)(1)=120
7!=(7)(6)(5)(4)(3)(2)(1)=5040
The Permutaion Formula is
nPr = n/(n-r)
An example is
n=7 and r=4
7p4= 7!/4!
Step one: Change your 7! to (7)(6)(5)(4)! That way, you can cancel with the 4! in the denominator.
So, you are left with (7)(6)(5) in the numerator, and just solve for that. Which should give you, 210.
The Combination formula is
nCr = n!/r!(n-r)!
[It is just the same as the permutations formula, with an extra r! in the denomenator]
An example of using this formula is:
n=7 r=4
7C4 = 7!/4!(7-4)!
= 7!/4!3!
do the same thing you did to the 7! in the numerator as you did in the last problem, and cancel both of the 4!
=(7)(6)(5)(4)!/4!3!
this time, change your 3! to (3)(2)(1), and cancel everything that you can in the numerator and denominator.
= (7)(6)(5)/(3)(2)(1)
= (7)(2)(5)/(2)
= 35
Another thing to note in this lesson is, in a permutation formula-if you see the word "and"-that is a big clue that you need to multiply!
In a combination formula-if you see the word "or"-that is a big clue that you need to add!
Lesson 6.7
Permutations and Combinations
With today's lesson you will learn how to solve Permutations and Combinations using their formulas.
Permutation is an arrangement of items in a particular order.
You can use the Multiplication Counting Principle or the Factorial Notation when you choose all of the items of a particular set.
Number of Permutations- n items of a set arranged r items at a time is nPr
nPr = n!
-----
(n-r)!
Example:
10P4= 10! 10!
----- = ---- = 5040
(10-4)! 6!
Combinations are a selection of items where the order does not matter.
You can solve Combinations using the formula:
nCr = n!
------
r!(n-r)!
Example:
5C3 = 5! 5! 120
------ = ---- = ----- = 10
3!(5-3)! 3!2! 6x2
**Remember** or means to add. And means to multiply.
Monday's Lesson was Chapter 6-7 on Permutations and Combinations.
Objective:Counting permutations and Combinations
permutation- an arrangement of items in a particular order.
Combination- a seolection in which order doesnt matter.
*Key Words when looking for a permutation are order and arrangement.
Formulas-
permutation:nPr=n!/(n-r)!
combination:nCr=n!/r!(n-r)!
Examples:
permutation-How many 6 letter codes can be made if no letter can be used twice?
10*10*26*26*26*26*25=1188137600
Combination-Evaluate 12C3
12!/3!9!=12*11*10*9/3*2 9!=220
This lesson's Botttom line:
permutation-
Both items are selected and the order of their selection are important
Combinations-
the only thing is which items are selected.
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