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Permutations. Warm Up – Find the Mean and the Standard Deviation. Warm Up – Find the Mean and Standard Deviation. Objective. Find Sample Space using Permutations and Combinations. Relevance.
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Objective • Find Sample Space using Permutations and Combinations
Relevance • Learn various methods of finding out how many possible outcomes of a probability experiment are possible. • Use this information to find probability.
Definition…… • Permutation – an arrangement of objects in a specific order. Order Matters!
How many ways can you arrange 3 people for a picture? Note: You are using all 3 people Answer: Example……
This is the same as using a factorial: Using the previous example: Factorial……
Example: Find 5! Calculator Steps: a. 5 b. Math c. Prb d. 4:! e. Enter Factorial can be found on our graphing calculator……
Suppose a business owner has a choice of 5 locations in which to establish her business. She decides to rank them from best to least according to certain criteria. How many different ways can she rank them? Answer: Note: She ranked ALL 5 locations. Example……
What if she only wanted to rank the top 3? Answer: This is no longer a factorial problem because you don’t rank ALL of them.
where n = total # of objects and r = how many you need. “n objects taken r at a time” Permutation Rule……
Remember the business woman who only wanted to rank the top 3 out of 5 places? This is a permutation:
Example: Find Calculator Steps: 5 Math Prb 2:nPr 3 Enter Permutations can be found on the graphing calculator……
A TV news director wishes to use 3 news stories on the evening news. She wants the top 3 news stories out of 8 possible. How many ways can the program be set up? Answer: Example……
How many ways can a chairperson and an assistant be selected for a project if there are 7 scientists available? Answer: Example……
How many different ways can I arrange 3 box cars selected from 8 to make a train? Answer: Example……
How many ways can 4 books be arrangedon a shelf if they can be selected from 9 books? Answer: Example……
How many ways can 4 books be arranged on a shelf? You can do 4! or you can set it up as a permutation. Answer: A factorial is also a permutation……
Note…… • 0! = 1 and 1! = 1
Order Words…… • How many ways can I • Listen • Sing • Read • 1st/2nd/etc • Pres/Vice-Pres • Chair/Assistant • Eat
Special Permutation when letters must repeat…… • Example: How many permutations of the word seem can be made? • Since there are 4 letters, the total possible ways is 4! IF each “e” is labeled differently. Also, there are 2! Ways to permute e1e2. But, since they are indistinguishable, these duplicates must be eliminated by dividing by 2!.
This leads to another permutation rule when some things repeat…… • It reads: the # of permutations of n objects in which k1 are alike, k2 are alike, etc.
Find the permutations of the word Mississippi. Number of Letters 11 – Total Letters 1 – M 4 – I 4 – S 2 - P Answer: You can eliminate the 1!’s because they are equal to 1. Example……
Definition…… • Combination – a selection of “n” objects withoutregard to order. Order Does NOT Matter!
Permutations of 2: AB CA AC CB AD CD BA DA BC DB BD DC Note: AB is NOT the same as BA. Combinations of 2: AB AC AD BC BD CD Note: AB is the same as BA Let’s compare ABCD – Find permutations of 2 and combinations of 2.
When different orderings of the same items are counted separately, we have a permutation problem, but when different orderings of the same items are not counted separately, we have a combination problem.
Combination Rule…… • Read: “n” objects taken “r” at a time.
How many combinations of 4 objects are there, taken 2 at a time? Answer: Example……
Find Calculator Steps: 4 Math Prb 3: nCr 2 Enter Combinations: There is a key on the graphing calculator……
To survey opinions of customers at local malls, a researcher decides to select 5 from 12. How many ways can this be done? Why is order is not important? Answer: Example……
A bike shop owner has 11 mountain bikes in the showroom. He wishes to select 5 to display at a show. How many ways can a group of 5 be selected? Note: He is NOT interested in a specific order. Answer: Example……
In a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there? The “and” indicates that you must use the multiplication rule along with the combination rule. Answer: Example……
In a club with 7 women and 5 men, select a committee of 5 with at least 3 women. This means you have 3 possibilities: 3W,2M or 4W,1M or 5W,0M Now you must use the multiplication rule as well as the addition rule. The reason for this is you are using “and” and “or.” Example……
Answer…… • 3W,2M: • 4W,1M: • 5W,0M: Add the totals: 350 + 175 + 21 = 546
In a club with 7 women and 5 men, select a committee of 5 with at most 2 women. This means you have 3 possibilities: 0W,5M or 1W,4M or 2W,3M Use the multiplication rule and the addition rule. First you multiply, then you add. Example……
Answer…… • 0W,5M: • 1W,4M: • 2W,3M: • Add the totals: 1 + 35 + 210 = 246
