7.5  Methods of Counting

 

Fundamental Counting Principle       

 

          If event M done m ways, followed by event N done n ways, then the event  can be done

m´ n ways.  Similar if more than 2 events 

 

1)    How many ways can you select a blouse and a skirt from 6 blouses, 4 skirts?

 

 

2)    How many answer sheets are possible from a 5 question, Sometimes/Always/Never test?

 

 

3)    If 20 people compete in a foot race, how many ways 1st, 2nd, 3rd occur?

 

 

4)    How many radio call numbers can be made with a K or W followed by 3 letters?

 

 

5)  If you are given a list of 5 TV shows and are asked to list your top 3 choices, how many ways could you do this?

 

 

Permutations of Unlike Objects

  **ALL objects are included

          If there are n objects, then there are n! ways to arrange them in a definite order with no repeats.

 

         

Counting Ordered Subsets  … permutations

 

8 Books on a shelf - how many ways can they be arranged?

 

Try smaller case of 3 books and LIST THEM -  

 

 

 

 

FACTORIALS    For whole number n,  

 

Find on calculator   5! =                                               0! = 1

 

 

 

 

 

 

In how many ways can the letters of SCRAMBLE be rearranged?

 

 

 

 

Permutation  -  If n different (UNLIKE) objects are chose r at a time, the number of possible permutations is

 

                            

 

 

Test on calculators               

 

 

 

Permutations implies Order without replacement

 

 

 

How many 4 letter "words" from “pickles” if no letter can be repeated?

 

 

 

From a set of 10 people - How many ways you choose President, V-Pres., Secretary, and Treasurer

 

Cross Country Team of 7 – How many ways can the top five finishers order be determine?

 

 

 

How many ways could you arrange 4 out of 9 ballerinas in a line?

 

 

 

 

A baseball team has 9 players.  Find the number of ways the manager can arrange the batting order.

 

 

 

 

Permutations Involving LIKE objects

 

          List the possible permutations from the letters in the word BED  versus the letters in BEE

 

 

 

 

 

Formula for permuting with Like objects

          If a set of n objects, of which r1 are one kind, r2 are of another kind, and so on through rk then the possible permutations of all n objects is

           

 

 

Find the number of ways the letters in the following words can be arranged:

BOOK                                               MISSISSIPPI                                           

 

 

 

BUBBLE                                           ALABAMA

 

 

 

In a car race, there are six Chevrolets, four Fords, and two Pontiacs.  In how many ways can the twelve cars finish if we consider only the makes of the cars? 

 

 

 

 

COMBINATION - order doesn't matter

 

 

CONTRAST Combinations versus Permutations.

Consider the set of magazines {Time, Life, People, NW}  

 

Choose two to take on a trip

Choose 1 to read today, and another for Sat

Combinations = sets

Permutations = Ordered Arrangements

{Time, Life} {Time, People} {Time, NW}

Time-Life, Time-People, Time-NW

{Life, People} {Life, NW}

Life-Time, Life-People, Life-NW

{ People, NW}

People-Time, People-Life, People-NW

   

 

 

For Combinations - ORDER IS NOT IMPORTANT, membership in set is

 

 

Formula for the number of Combinations of n objects taking r at a time

                                                             

                   

                  

                                                                                  

 

 

 

 

In a class of 35 kids a teacher needs a group of three kids to run errand, how many ways can choose which three?

 

If you take 5 points in a plane, no three of them on a line, how many straight lines can be drawn if each line is drawn through a pair of points?

 

 

 

 

How many different 5 card hands from deck of 52?

 

 

 

The FDA is testing a new drug.  18 people volunteer to take the new drug.  5 of the volunteers receive a placebo, and other 13 get the drug. 

How many different placebo groups are possible? 

 

How many drug groups are possible?

 

 

 

 

Using Counting Techniques in Probability

 

There are 52 - 7th grade girls on overnight field trip - principal randomly assigns 4 girls/room

How many possible groups of 4?

 

 

What is probability particular group of 4 will be together?

 

 

Beth & Kate want together.  What is the probability they will be?

 

 

Allison, Erin, & Sara want together.  What is probability this occurs?

 

 

Stacey & Tracy don't want together.  What is probability they are not? 

 

 

 

 

In a six question TF test, what is the probability that a student gets exactly 50%?

          Note – tree diagram     versus  probability with permutations

 

 

 

 

 

 

How many different ways can letters M, A,T, H be arranged?

 

What is probability they spell MATH?

 

 

 

 

A high school student must take Math, English, P.E., Science, History & Language & lunch and Study Hall.  Everyone's lunch is 5th hour.  Day has 8 periods.  How many possible schedules?

 

 

 

 

 

Social Security Number -  _ _ _ - _ _ - _ _ _ _    How many possible?

 

 

 

 

License Plates   Letter-Letter-Digit-Digit-Digit

How many different license plates are possible?

 

What is Probability  RJ123?

 

What is probability RJ _ _?

 

 

 

 

Ten Question T/F, what is probability of guessing and getting 60%