home

=Combinations:Welcome to our class Wiki! = *PLEASE CITE YOUR MATERIAL GRADES:

PROBABILITY PROJECT
-Permutations are used in alot of things like Geometry, Algebra, Probability, number of theory and alot more. Around 1770 was when the first mathematical permutations were used by Joesph Louis Lagrange. (n.d.). Retrieved from []
 * History of Permutations.**

1. The number of ways a multi-part task can occur equals the prod​uct of the number of ways to complete each independent part of the task. "If there are r ways to do one thing, and s ways to do another, and t ways to do a third thing, and so on ..., then the number of ways of doing all those things at once is r x s x t x ..." (n.d.). Retrieved from [] 2. Different Methods - -The **factorial function** (symbol: **!**) Just means a series of numbers that decline down to one. Example: 4x3x2x1=24 or 4!=24 (n.d.). Retrieved from []
 * ​I. Fundamental Counting Principles**
 * ​ //For Example:// **
 * __Factorials:__ Factorials are used to abbreviate products that start at one number and work their way back to 1 by using an exclamation point (!). Ex: 5! = 5 x 4 x 3 x 2 x 1 = 120

8 //P//  3  can be expressed in terms of factorials as ( 8 − 3 )! || = || __8!__ 5! || (n.d.). Retrieved from [] ex. Tree Diagram- > **Repetition is allowed: As a lock the code could be 444. > No Repetition: As an example the first three people in a running race, you cannot be First and Second at the same time **.
 * 8 //P//  3  || = || __ 8 !__
 * __Tree Diagrams:__ Tree Diagrams are used to compare the number of possible outcomes for each situation. By listing the situations and the possible outcomes we can determine the relationship between the number of decisions and the number of outcomes.
 * __Probability:__ Probability is used to determine the probability of an event to occur. This is defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space.
 * __Odds:__ Odds are used to determine the odds in favor and the odds against. (when all outcomes are equally likely) (n.d.). McDougal Littel Algebra 2 math book
 * __Another Example__: A license plate consisting of three letters and three numbers.

(n.d.). Retrieved from [] > How many different license plates are there altogether?

LETTER LETTER LETTER NUMBER NUMBER NUMBER

For each of the letters we have 26 choices. For each of the numbers we have 10 choices.

26 || 26 || 26 || 10 || 10 || 10 ||
 * The number of ways to pick the first letter || The number of ways to pick the second letter || The number of ways to pick the third letter || The number of ways to pick the first number || The number of ways to pick the second number || The number of ways to pick the third number ||

The Fundamental Counting Principle says that: //The total number of ways to fill the six spaces on a licence plate is which equals 17,576,000// (n.d.). Retrieved from []
 * 26** x **26** x **26** x **10** x **10** x **10**

(n.d.). Retrieved from http://www.fallingfifth.com/files/comics/factorial.png

1. Ordered Combination -an ordered arrangement of the numbers, terms, etc., of a set into specified groups -A permutation is one of the different classification of a batch of items were order always matters. (n.d.). Retrieved from http://www.omegamath.com/Data/d2.2.html - Permutations is when you change the order of elements arranged in a particular order, as // def // into // dfe, efd, // etc., or of arranging a number of elements in groups made up of equal numbers of the elements in different orders, as // d // and // e // in // de // and // ed. //  (n.d). Retrieved From [] ​ ***To help you to remember, think "Permutation ... Position"***
 * ll. Permutations**

1. Permutations with Repetition
These are the easiest to calculate. When you have //n// things to choose from ... you have //n// choices each time! So when choosing //r// of them, the permutations are: n × n × ... (r times) = nr (Because there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on.) Example: in the lock above, there are 10 numbers to choose from (0,1,..9) and you choose 3 of them: 10 × 10 × ... (3 times) = 103 = 1000 permutations. (n.d.). Retrieved from [] 2. P(n, r)= nPr=N! / (N-R)! 3. We can use it when order matters. 4. We can use this to count how many items a person needs. This is used in adjutant jobs, such as in the shipping industry and the like. 5. __Real life examples:__ the arrangement of books on a shelf & the placing of 8 different marbles in 8 different boxes. The lottery would be another example. Ex 1: The lottery is a very large game of chance. You can win big if you get the right combination of numbers. Example 2:

By The Permutations of the letters //abc// we mean all of their possible arrangements:

abc acb bac bca cab cba

There are 6 permutations of three different things. (n.d.). Retrieved from [] How many ways can first and second place be awarded to 10 people?

 3360 Ways. (n.d.). Retrieved from http://www.mathsisfun.com/combinatorics/combinations-permutations.html

6. nPr; n=6 and r=24, nPr; n=12 r=4, nPr; n=4 r=6 **1)** There are 6! permutations of the 6 letters of the word //square//. a) In how many of them is //r// the second letter? _ //__r__// _ _ _ _ b) In how many of them are //q// and //e// next to each other? (n.d.). Retrieved from []

**2)** Suppose you have forgotten you locker combination. There are 36 numbers on the lock, and the correct combination order is right __- left__ - right __. How many combinations may be necessary to try in order to open the lock? (Hint: Numbers such as 3-3-3 may be repeated)__

(n.d.). Retrieved from []

__3)__ 5!__ || __=__ || __8**·** 7**·** 6__ || ​
 * example of factorials:
 * __8!

4)BOB MIKE SUE ALICE Given 4 people, Ryan, Lanse, John and Mike. How many different ways can these four people be arranged where order matters? If we find all possible arrangements of Ryan, Lanse, John and Mike where order matters, we have the following: RLJM, RLMJ, RJLM, RJML, RMLJ, RMJL LRJM, LRMJ, LMRJ, LMJR, LJRM, LJMR JRLM, JRML, JLRM, JLMR, JMRL, JMLR MRLJ, MRJL, MLRJ, MLJR, MJRL, MJLR There are 24 ways to arrange the four people four at a time, or 4! (n.d.). Retrieved from [] 5)Find the number of ways to arrange 6 items in groups of 4 at a time where order matters.

//Solution:// //There are 360 ways to arrange 6 items taken 4 at a time when order matters// []

1. Combinations are collections of things, in which the order does not matter. <span style="color: #fb600e; font-family: 'Lucida Console',Monaco,monospace;">2. Notation: nCr= n!/ r!(n-r)! <span style="background-color: #ffffff; color: #00b1ff; font-family: 'Lucida Console',Monaco,monospace;">3. We use combinations when trying to figure out how many things can be arranged. <span style="background-color: #ffffff; color: #000000; font-family: 'Lucida Console',Monaco,monospace;"> **(** [|**http://www.mathsisfun.com/combinatorics/combinations-permutations.html**] **).** <span style="color: #000000; font-family: 'Lucida Console',Monaco,monospace;">**Ex:** Five people are in a club and three are going to be in the 'planning committee,' to determine how many different ways this committee can be created we use our combination formula as follows: 5C3= 5!/ 3!(5-3)! 5*4*3*2*1/ 3*2*1(2)! 5*4*3*2*1/ 3*2*1*2*1(cross out the 3 & 2 on top & bottom) left over is: 5*4/ 2*1= 10. ([])
 * III. Combinations**

4. Combinations help us find all the different ways to arrange something. [] <span style="color: #ff00a3; font-family: Georgia,serif;"> 5) If the order __doesnt__ matter, then its a combination. If the order __does__ matter, then its a permutation. <span style="color: #000000; font-family: 'Lucida Console',Monaco,monospace;">[]

6) The N stands for the number of things you can choose from and the r stands for repition. []

<span style="color: #00ff44; font-family: 'Comic Sans MS',cursive; font-size: 130%;">7) If something can be chosen, or can happen, or be done, in m different ways, and, after that has happened, something else can be chosen in n different ways, then the number of ways of choosing both of them is m · n. [] <span style="color: #000000; font-family: 'Lucida Console',Monaco,monospace;"> 8) __Combinations:__ a un-ordered collection of distinct elements.( [|http://www.google.com/search?]  <span style="background-color: #ffffff; color: #000000; font-family: 'Lucida Console',Monaco,monospace;"> IV. Pascal's Triangle <span style="color: #000000; font-family: 'Lucida Console',Monaco,monospace;">1. Each line in Pascal's Triangle is formed by adding together each pair of adjacent numbers in the line above.

** 3)Pascals triangle is a well known mathematical pattern. This pattern was actually discovered in China, but it has been named after the first westerner to study it. ** [] <span style="color: #e60f11; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 110%;">4) Pascals triangle plays an important role in binomial theorem, raise of powers and combinations [] <span style="color: #ff00a3; font-family: 'Times New Roman',Times,serif;">** 5) B ** ([])
 * Patterns within Pascals Triangle **
 * The first diagonal is, of course, just "1"s, and the next diagonal has the counting numbers. ( 1,2, 3 etc.) **
 * The third Diagnol has the triangular numbers. **
 * The fourth Diagnol has the tetrahedral numbers. **

=
<span style="color: #ffb600; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 110%;">Like so many other branches of mathematics, the development of probability theory has been stimulated by the variety of its applications. Conversely, each advance in the theory has enlarged the scope of its influence. Mathematical statistics is one important branch of applied probability; other applications occur in such widely different fields as genetics, psychology, economics, and engineering. Many workers have contributed to the theory since Laplace's time; among the most important are Chebyshev, Markov, von Mises, and Kolmogorov. ======

=
<span style="color: #ffc300; font-family: 'Lucida Sans Unicode','Lucida Grande',sans-serif; font-size: 110%;">In 1933 a monograph by a Russian mathematician A. Kolmogorov outlined an axiomatic approach that forms the basis for the modern theory. [] ======

[|http://www.cc.gatech.edu/classes/cs6751_97_winter/Topics/stat-meas/probHist.html)B] <span style="background-color: #00fffa; font-family: Tahoma,Geneva,sans-serif;">[|**http://www.probabilitytheory.info/topics/pascal_combinations_permutations.htm**]) __ A real life example of combinations is picking nominees for a council, or picking names out of a hat, and figuring out the different ways they can be drawn. Retrieved from ( []) ​ **Problem 1)** In a class of 10 students, how many ways can a club of 4 students be arranged? 10C4= 210 . (__[]__) 1) <span style="color: #0000ff; font-family: 'Comic Sans MS',cursive;">**10 C 4= 10!/(10-4)!*4!= 210.** 2) **<span style="color: #32ff00; font-family: 'Comic Sans MS',cursive;">3 C 2=3. ** 3) **<span style="color: #fb600e; font-family: 'Comic Sans MS',cursive;">8 C 6=28. ** 5A .The pascal triangle was named after Blaise Pascal, who was a famous french mathematician and philosopher. __ __(__[]__)__
 * Problems to solve.**

5A) This drawing is entitled "The Old Method Chart of the Seven Multiplying Squares ". <span style="color: #ff00a3; font-family: 'Times New Roman',Times,serif; font-size: 120%;">It is from the front of a book called "//Ssu Yuan Yü Chien" (Precious Mirror of the Four Elements)//, written in AD 1303 and in the book it says the triangle was known about more than two centuries before that. ([])

__ 5B .Pascal triangles are useful in Algebra and in probability .([]) EX (x+1)^0 = 1 (x+1)^1 = 1 + x (x+1)^2 = 1 + 2x + x^2 (x+1)^3 = 1 + 3x + 3x^2 + x^3 (x+1)^4 = 1 + 4x + 6x^2 + 4x^3 + x^4 (x+1)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5 ..... ([]) EX 6A .They can be used if you have 10 hat on a rack and you want to find out how many diffrent way you can take two hats with you. ( []) EX 6B .They can be used if you have if you went to the store and wanted to get 4 shirts but there where 30 to pick from you can see how many ways you could pick 4 shirts from 30.(__ []__)__

1. There were 20 people that wanted to be in a club and there were only 6 spots. How many different combinations can there be?

2. There are 15 people in a race. How many different ways can people get 1st,2nd,and 3rd?

3. There are 52 cards in a deck of cards. What is your chanse of picking a face card?

[] IV [] http://www.mathagonyaunt.co.uk/STATISTICS/ESP/Perms_combs.html
 * <span style="background-color: #32ff00; font-family: 'Arial Black',Gadget,sans-serif; text-decoration: line-through;">__IV: The difference between permutation and combinations, is combinations order doesn't matter and permuation, order does matter. (__ **[|**http://www.mathsisfun.com/combinatorics/combinations-permutations.html**]**<span style="background-color: #32ff00; font-family: 'Arial Black',Gadget,sans-serif; text-decoration: line-through;">__)__ **
 * <span style="font-family: Tahoma,Geneva,sans-serif; font-size: 110%;">the first book of probability, De Ratiociniis in Ludo Aleae, was published in 1657.
 * <span style="color: #000000; font-family: 'Arial Black',Gadget,sans-serif;">Permutations : Position important.
 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Combination's : Chosen important.
 * there are specific placement requirements
 * repetition is permitted


 * a visual representation of all the possible outcomes of a multi-part task.
 * use the fundamental counting principle to determine the total number of outcomes and then set up the tree diagram using each part of the task as headings. List all then possible outcomes under each heading. Each possible outcome from the first heading will create a branch to each possible outcome from the next heading.


 * ex: Aidan is playing a game where he flips a coin and then rolls a dies. construct a tree diagram and list the outcomes in a sample space.

heads: 1,2,3,4,5,6 Taks 2 tails: 1,2,3,4,5,6


 * Problem: || A spinner has 4 equal sectors colored yellow, blue, green and red. What are the chances of landing on blue after spinning the spinner? What are the chances of landing on red? || [[image:http://www.mathgoodies.com/lessons/vol6/images/tab.gif width="15" height="1" caption=" "]][[image:http://www.mathgoodies.com/lessons/vol6/images/spinner-0.gif width="80" height="80" caption="spinner"]] ||
 * Solution: || The chances of landing on blue are 1 in 4, or one fourth. ||^  ||
 * || The chances of landing on red are 1 in 4, or one fourth. ||^  ||


 * **Definition** || **Example** ||
 * An experiment is a situation involving chance or probability that leads to results called outcomes. || In the problem above, the experiment is spinning the spinner. ||
 * An outcome is the result of a single trial of an experiment. || The possible outcomes are landing on yellow, blue, green or red. ||
 * An event is one or more outcomes of an experiment. || One event of this experiment is landing on blue. ||
 * Probability is the measure of how likely an event is. || The probability of landing on blue is one fourth. ||


 * **Probability Of An Event** ||
 * || P(A) = || The Number Of Ways Event A Can Occur ||
 * ^  || The total number Of Possible Outcomes ||   ||


 * Experiment 1: || A spinner has 4 equal sectors colored yellow, blue, green and red. After spinning the spinner, what is the probability of landing on each color? || [[image:http://www.mathgoodies.com/lessons/vol6/images/tab.gif width="10" height="1" caption=" "]] || [[image:http://www.mathgoodies.com/lessons/vol6/images/spinner-0.gif width="80" height="80" caption="spinner"]] ||
 * Outcomes: || The possible outcomes of this experiment are yellow, blue, green, and red. ||^  ||^   ||
 * Probabilities: ||  || P(yellow) || = = || ## of ways to land on yellow ||   || 1 ||
 * ^  ||^   || total # of colors ||^   || 4 ||
 * P(blue) || = = || ## of ways to land on blue ||  || 1 ||
 * ^  ||^   || total # of colors ||^   || 4 ||
 * P(green) || = = || ## of ways to land on green ||  || 1 ||
 * ^  ||^   || total # of colors ||^   || 4 ||
 * P(red) || = = || ## of ways to land on red ||  || 1 ||
 * ^  ||^   || total # of colors ||^   || 4 ||   ||
 * P(red) || = = || ## of ways to land on red ||  || 1 ||
 * ^  ||^   || total # of colors ||^   || 4 ||   ||
 * ^  ||^   || total # of colors ||^   || 4 ||   ||


 * Experiment 2: || A single 6-sided die is rolled. What is the probability of each outcome? What is the probability of rolling an even number? of rolling an odd number? || [[image:http://www.mathgoodies.com/lessons/vol6/images/tab.gif width="10" height="1" caption=" "]][[image:http://www.mathgoodies.com/lessons/vol6/images/dice.gif width="220" height="64" caption="dice"]] ||
 * Outcomes: || The possible outcomes of this experiment are 1, 2, 3, 4, 5 and 6. ||^  ||
 * Probabilities: ||||  ||   || P(1) || = = || ## of ways to roll a 1 ||   || 1 ||
 * ^  || total # of sides || 6 ||^   ||
 * P(2) || = = || ## of ways to roll a 2 ||  || 1 ||
 * ^  ||^   || total # of sides ||^   || 6 ||
 * P(3) || = = || ## of ways to roll a 3 ||  || 1 ||
 * ^  ||^   || total # of sides ||^   || 6 ||
 * P(4) || = = || ## of ways to roll a 4 ||  || 1 ||
 * ^  ||^   || total # of sides ||^   || 6 ||
 * P(5) || = = || ## of ways to roll a 5 ||  || 1 ||
 * ^  ||^   || total # of sides ||^   || 6 ||
 * P(6) || = = || ## of ways to roll a 6 ||  || 1 ||
 * ^  ||^   || total # of sides ||^   || 6 ||
 * P(even) || = = || ## ways to roll an even number ||  || 3 || = || 1 ||
 * ^  ||^   || total # of sides ||^   || 6 ||^   || 2 ||
 * P(odd) || = = || ## ways to roll an odd number ||  || 3 || = || 1 ||
 * ^  ||^   || total # of sides ||^   || 6 ||^   || 2 ||   ||   ||
 * ^  ||^   || total # of sides ||^   || 6 ||
 * P(even) || = = || ## ways to roll an even number ||  || 3 || = || 1 ||
 * ^  ||^   || total # of sides ||^   || 6 ||^   || 2 ||
 * P(odd) || = = || ## ways to roll an odd number ||  || 3 || = || 1 ||
 * ^  ||^   || total # of sides ||^   || 6 ||^   || 2 ||   ||   ||
 * P(odd) || = = || ## ways to roll an odd number ||  || 3 || = || 1 ||
 * ^  ||^   || total # of sides ||^   || 6 ||^   || 2 ||   ||   ||

[|equally likely]
 * Experiment 3: || A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing a red marble? a green marble? a blue marble? a yellow marble? || [[image:http://www.mathgoodies.com/lessons/vol6/images/tab.gif width="10" height="1" caption=" "]] || [[image:http://www.mathgoodies.com/lessons/vol6/images/marbles_jar.gif width="100" height="100" caption="[IMAGE]"]] ||
 * Outcomes: || The possible outcomes of this experiment are red, green, blue and yellow. ||^  ||^   ||
 * Probabilities: ||||  || P(red) || = = || ## of ways to choose red ||   || 6 || = || 3 ||
 * ^  || total # of marbles || 22 ||^   || 11 ||^   ||
 * P(green) || = = || ## of ways to choose green ||  || 5 ||
 * ^  ||^   || total # of marbles ||^   || 22 ||
 * P(blue) || = = || ## of ways to choose blue ||  || 8 || = || 4 ||
 * ^  ||^   || total # of marbles ||^   || 22 ||^   || 11 ||
 * P(yellow) || = = || ## of ways to choose yellow ||  || 3 ||
 * ^  ||^   || total # of marbles ||^   || 22 ||   ||
 * P(yellow) || = = || ## of ways to choose yellow ||  || 3 ||
 * ^  ||^   || total # of marbles ||^   || 22 ||   ||
 * ^  ||^   || total # of marbles ||^   || 22 ||   ||


 * Experiment 4: || Choose a number at random from 1 to 5. What is the probability of each outcome? What is the probability that the number chosen is even? What is the probability that the number chosen is odd? ||
 * Outcomes: || The possible outcomes of this experiment are 1, 2, 3, 4 and 5. ||
 * Probabilities: ||  || P(1) || = = || ## of ways to choose a 1 ||   || 1 ||
 * ^  ||^   || total # of numbers ||^   || 5 ||
 * P(2) || = = || ## of ways to choose a 2 ||  || 1 ||
 * ^  ||^   || total # of numbers ||^   || 5 ||
 * P(3) || = = || ## of ways to choose a 3 ||  || 1 ||
 * ^  ||^   || total # of numbers ||^   || 5 ||
 * P(4) || = = || ## of ways to choose a 4 ||  || 1 ||
 * ^  ||^   || total # of numbers ||^   || 5 ||
 * P(5) || = = || ## of ways to choose a 5 ||  || 1 ||
 * ^  ||^   || total # of numbers ||^   || 5 ||
 * P(even) || = = || ## of ways to choose an even number ||  || 2 ||
 * ^  ||^   || total # of numbers ||^   || 5 ||
 * P(odd) || = = || ## of ways to choose an odd number ||  || 3__ ||
 * ^  ||^   || total # of numbers ||^   || 5 ||   ||
 * P(even) || = = || ## of ways to choose an even number ||  || 2 ||
 * ^  ||^   || total # of numbers ||^   || 5 ||
 * P(odd) || = = || ## of ways to choose an odd number ||  || 3__ ||
 * ^  ||^   || total # of numbers ||^   || 5 ||   ||
 * P(odd) || = = || ## of ways to choose an odd number ||  || 3__ ||
 * ^  ||^   || total # of numbers ||^   || 5 ||   ||


 * Summary: || The probability of an event is the measure of the chance that the event will occur as a result of an experiment. The probability of an event A is the number of ways event A can occur divided by the total number of possible outcomes. The probability of an event A, symbolized by P(A), is a number between 0 and 1, inclusive, that measures the likelihood of an event in the following way:* If P(A) > P(B) then event A is more likely to occur than event B.
 * If P(A) = P(B) then events A and B are equally likely to occur. ||
 * Other notable things about probability: if the order does NOT matter, it is a combination. If the order DOES matter, then it is a permutation. According to www,mathisfun,com and www.mathgoodies.com, when solving a permutation problem, it is sometimes easier to use notations such as this one[[image:http://www.mathsisfun.com/combinatorics/images/permutation-notation.png]] ..... instead of writing out the entire formula.
 * Probability can be very confusing. It can stump the best of our brains, and make us pull our hair out. But understanding the simple little things about probability, will help you catch on to things much faster. Like the formulas, if you know how to do the formulas (which are very easy by the way) then theres a GIANT first step on understanding probability. Even after awhile, the god forbidden binomial expansion gets easy. You just have to concentrate. Anchor down on learning, and focus, and you will understand it in a matter of minuets.

[] [|http://www.ucc.edu/NR/rdonlyres/D55520DE-CD3A-42CB-8CEE-471329AA6A91/0/CountingMethods.pdf]
 * ** Some methods to count the number of ways items can be arranged are factorials, permutations and combinations. **

EXAMPLE: <span style="font-family: Georgia,serif; font-size: 110%;">If a dice is rolled once, what is the probability that it will show a multiple of 3 ? 2:6 [] <span style="background-color: #fb780e; color: #00fffa; font-family: Arial,Helvetica,sans-serif;">

Combinations: 1.) Order 2.)nCk //**combination notation**//. [] 3.) Arrangements [] 4.) it helps alot with jobs and the yearly count of people [] 5.)arrangement [|**binomial coefficients**] in a [|triangle] . a.)It is named after the French mathematician [|Blaise Pascal] in much of the [|Western worl] b.)



6.) a.) Example: in the lock above, there are 10 numbers to choose from (0,1,..9) and you choose 3 of them: **10 × 10 × ... (3 times) = 103 = 1000 permutations** []

b.)Another example: 4 things can be placed in **4!** **= 4 × 3 × 2 × 1 = 24** different ways

[]

7.) 8.) The difference between permutations and combinations is that permutations are not needed in order but combinations are. 9.) A.) []

B.)

C.) D.)http://images.google.com/images?hl=en&tbo=1&gbv=2&tbs=isch:1&ei=2wrFS9-bMI7QNJCQ6OkN&sa=X&oi=spell&resnum=0&ct=result&cd=1&q=combinations+and+permutations&spell=1

10) an example of what a combination problem can look like is this- A basketball team has 11 players on its roster. Only 5 players can be on the court at one time. How many different groups of 5 players can the team put on the floor?

^ you can put them in any order, you just need to find how many groups you can make with the numbers they give you. []