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 Learn various methods of finding out how many possible outcomes of a probability experiment are possible. Use this information to find probability.

Definition…… Order Matters! Permutation – an arrangement of objects in a specific order. Order Matters!

Example…… How many ways can you arrange 3 people for a picture? Note: You are using all 3 people Answer:

Factorial…… This is the same as using a factorial: Using the previous example:

Factorial can be found on our graphing calculator…… Example: Find 5! Calculator Steps: a. 5 b. Math c. Prb d. 4:! e. Enter

Example…… 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.

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.

Permutation Rule…… where n = total # of objects and r = how many you need. “n objects taken r at a time”

Remember the business woman who only wanted to rank the top 3 out of 5 places? This is a permutation:

Permutations can be found on the graphing calculator…… Example: Find Calculator Steps: 5 Math Prb 2:nPr 3 Enter

Example…… 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 arranged on a shelf if they can be selected from 9 books? Answer:

A factorial is also a permutation…… 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:

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!.

How many permutations of the word seem can be made? Answer:

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.

Example…… 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.

Combinations

Definition…… Order Does NOT Matter! Combination – a selection of “n” objects without regard to order. Order Does NOT Matter!

Let’s compare ABCD – Find permutations of 2 and combinations 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

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.

Example…… How many combinations of 4 objects are there, taken 2 at a time? Answer:

Combinations: There is a key on the graphing calculator…… Find Calculator Steps: 4 Math Prb 3: nCr 2 Enter

Example…… 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.”

Answer…… 3W,2M: 4W,1M: 5W,0M: Add the totals: 350 + 175 + 21 = 546

Example…… Use the multiplication rule and the addition rule. 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.

Answer…… 0W,5M: 1W,4M: 2W,3M: Add the totals: 1 + 35 + 210 = 246