How Many is Too Many?

A counting principle investigation

A tree diagram is a graphic organizer used to list all possibilities of a sequence of events in a systematic way.  Tree diagrams are one method for calculating the total number of outcomes in a sample space.

The senior class is planning a trip to the Virgin Islands.  The class officers are exploring various travel options.  From the high school, they will travel to Miami either by car ,bus ,train or plane. To travel to St. Thomas from Miami, they can take a plane or a ship . Draw a tree diagram to clearly illustrates each of the possible ways the group can travel on their trip. 

#1

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Answer to previous problem

Three coins are tossed: a dime, a nickel and a penny.  Construct a tree diagram showing all of the possible outcomes for flipping the three coins in an organized manner. 

#2

Construct a tree diagram to illustrate the different single-dip ice cream sundaes can be made from the ice cream flavors: chocolate, vanilla, and strawberry and the topping choices: fudge and butterscotch

#3

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Your Cone

The Fundamental Counting Principle states that

if event M can occur in m ways and event N can occur in n ways, then event M followed by

event N can occur in m · n ways.  

If event M can occur in 3 ways and event N can occur in 4 ways, then M followed by N can

occur in 3 ·4 or 12 ways.

#1 A nurse has three patients to visit. How many different ways can she make her rounds if she checks each patient once? 

#2 High school faculty are to be issued special coded identification cards that consist of four letters of the alphabet.   How many different ID cards can be issued if the letters can be used more than once?

Practice

#3 Suppose now that the letters in the high school identification codes cannot be repeated, how many different identification cards can be created?

                                                       #4 The digits 0, 1, 2, 3, and 4 are to be used in a 4-digit ID card.  How many different cards are possible if repetitions are permitted?

#5 The digits 0, 1, 2, 3, and 4 are to be used in a 4-digit ID card.  How many different cards are possible if repetitions are not permitted? 

#6 Kayla is in charge of a talent show at BJHS.  There are eight contestants.  She must choose in what order they will appear.  How many different ways can she schedule the performers? 

Answer to the nurse's problem:                                                     

For her first visit, she could visit any of the 3 patients.

For her second visit, she can visit either of the 2 remaining patients.

On her last visit, there will be only one patient left to see.

 The number of different ways can she can make her rounds is:

(1st visit)(2nd visit)( 3rd visit) =3 x 2 x 1 = 6

                                                    

Answer to the high school faculty problem(with alphabetic repetitions):

                                                    Because four letters are to be used, there are four

slots (positions) to fill.Any of the 26 letters of the alphabet can be used in each of the four code positions, therefore the total number of identification cards that can be

made is:(26)(26)(26)(26)=456,976

Answer to the high school faculty problem(without alphabetic repetitions):

                                                    Because the letters cannot be repeated, once a letter is used, it cannot be used

again.  Therefore, there will be 26 options for the first position, 25 options for the second position, 24 for the third, and 23

for the fourth.   (26)(25)(24)(23)=358,800

Answer to the 4-digit I.D. problem(with repetitions):

                                                     Four slots will be needed. Each slot will have five digits that could be used. The total number of identification cards that

can be made is(5)(5)(5)(5)=625

                                                     

Answer to the 4-digit I.D. problem(without repetitions):

                                                    Four slots will be needed. Each slot will have five digits that could be used. The total number of identification cards that

can be made is(5)(4)(3)(2)=120

Answer to Kayla's problem:                                                      

Kayla has eight contestants who can perform first, seven remaining

contestants who can perform second, six contestants who can perform third,

etc., etc.

There are eight slots (positions) to be filled.  The number of ways she can

select the order is illustrated as follows:   (8)(7)(6)(5)(4)(3)(2)(1)=40,320

Kayla can schedule the performances in 40,320 different ways!!