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Systematic Counting, Using Tree Diagrams to SolFundamental Counting Principle Counting Problems, Counting Techniques, Factorials. Permutations, and Combinations,

 

 

Questions on ­ Systematic Counting, Using Tree Diagrams to SolFundamental Counting Principleve
Counting Problems, Counting Techniques, Factorials. Permutations, and Combinations,
1. List all the ways to select two members from S with repetition. The order in which the members are selected is not important. For example, DD is allowed and BA is the same selection as AB.
S= {Q, R, S, T, U}. Identify the list of all the ways to select two members from S, allowing for repetition. Choose the correct answer below.
A. QQ, QR, QS, QT, QU, RR, RS, RT, RU, SS, ST, SU, TT, TU, UU
B. QS, QT, QU, RS, RT, RU, ST, SU
C.QQ, QR, QS, QT QU, RS, RT, RU, ST, SU, TU
D. QQ, QR, QS,QT,QU,RQ ,RR , RS, RT, RU,SQ ,SR ,SS ,ST,SU,TQ ,TR ,TS , TT ,TU 2. List all the ways to select two different members from S with repetition. The order in which members are selected is important. For example, AA is allowed and AB is not the same selection as BA.
S={B, C, D, E, F} Identify the list of all the ways to select two members from S. Choose the correct answer below.
a.BB, BC, BD, BE, BF, CC, CD, CE, CF, DD, DE, DF, EE, EF, FF
b..BD , BE, BF,CD,CE,CF,DE , DF
c..BB, BC, BD, BE, BF, CB, CC, CD, CE, CF, DB, DC, DD, DE, DF, EB, EC, ED, EE, EF FB, FC, FD, FE,
FF
d. BB, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF,
3. How many different two­digit numbers can you form using the digits 7, 2, 6, 8, 5, 4, 9, and 3 without repetition? For example, 88 is not allowed.
The number of different two­digit numbers =
4. How many different two­digit numbers can be formed using the digits 8, 3, 5, 1, 4, and 7 with repetition? For example, 33 is allowed.
The number of different two­digit numbers =
5. How many different three­digit numbers can be formed using the digits 3, 5, 1, 7, and 2 without repetition? For example, 511 is not allowed.
The number of different three­digit numbers =
6. Assume two five­sided dice are rolled each with the numbers 1, 2, 3, 4, and 5. The first one is red and the second one is green. Use a systematic listing to determine the number of ways to roll both numbers that are the same.
Use ordered pairs of the form (red number, green number) to represent the pairs showing on the dice. Choose the correct answer below.
a. (1, 2), (1, 3), (1, 4), (1, 5)
b. (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) c. (1,2), (1,3), (1,4), (3,1), (3,2), (3,4,), (4,1), (4,2), (4,3)
d. (1, 1), (2, 2), (3, 3), (5, 5) 7. Assume you are rolling two dice; the first one is red, and the second is green. Use a systematic listing to determine the number of ways you can roll a total less than 11 on the two dice.
In how many ways can you roll less than 11? ways 8. A designer has designed different tops, pants, and jackets to create outfits. How many different outfits can the models wear if she has designed the following pieces?
nine tops, two pants, eight jackets. There are a total of different outfits. 9. Susan's friends Vicki, Luis, Maddy, Hank, Rob, Sean, Marie, Roy and Jeanne have volunteered to help
with the preparations for a party. How many ways can Susan assign someone to buy beverages, someone to arrange for food, and someone to send the invitations? Assume that no person does two jobs. How many ways can Susan assign her volunteers? ways 10. If you drew a tree diagram showing how many ways 4 coins could be flipped, how many branches would it have? The tree diagram would have branches 11. An eyewitness to a crime said that the license plate of the getaway car began with the letters U, C, X, and A, but he could not remember the order. The rest of the plate had the numbers 6, 9, 8, and 3
but again he did not recall the order. How many license plates must the police investigate to find the car?
How many license plates must the police investigate to find the car? Choose the correct answer below.
A. 576
B. 40,320
C.16
D.256 12. Two couples (Adam/Brenda and Carl/Darlene) have bought tickets for four adjacent seats to see a musical. In how many ways can the couples be seated if the couples are to sit together?
There are ways 13. The board of an Internet startup company has seven members. If one person is to be in charge of customer relations and another in charge of marketing, in how many ways can these two positions be filled? How many ways can the two positions be filled?
14. An equestrian club has fourteen members. If the club wants to select a president, vice president, and treasurer (all of whom must be different), in how many ways can this be done?
How many ways can the three positions be filled? 15.The owner of a stereo store wants to advertise that he has many different sound systems in stock. The
store carries 7 different CD players, 9 different receivers, and 6 different speakers. Assuming a sound system consists of one of each, how many different sound systems can he advertise?
How many different sound systems can the owner advertise? 16. A game uses a 10­sided die. Determine the number of possibilities of rolling the 10­sided die twice. There are possibilities of rolling a 10­sided die twice. 17. Using the digits 0, 1, 2, ...8, 9, determine how many 7­digit numbers can be constructed according to
the following criteria. One cannot be used for the first digit; digits may be repeated.
The number of 7­digit numbers that can be constructed =
18. You are taking a multiple­choice test that has 6 questions. Each of the questions has 5 answer choices, with one correct answer per question. If you select one of these choices for each question and leave nothing blank, in how many ways can you answer the questions?
You can answer the questions in ways 19.  A college debate team is participating in a tournament in state A. They will fly from state B to state C, then to state D, and finally to state A. If there are 4 possible flights to state C, 3 from state C to state D,
and finally 6 from state D to state A, in how many ways can they schedule the trip?
How many ways can they schedule the trip? 20. In an experiment on social interaction, 5 people will sit in 5 seats in a row. In how many ways can this be done? There are ways 21. Simplify. (10−4)! (10−4)! = 22. Evaluate.15! 12! The solution is 23. Evaluate the expression (5,2). P(5,2)=
24. Evaluate the expression C(12,7). C(12,7)=..
25. Find the number of permutations. Nine objects taken six at a time
There are permutations 26. Find the number of combinations.11 objects taken 4 at a time. There are Combinations. 27. Specify the number of ways to perform the task described. Give your answers using P(n, r) or C(n, r) notation .Mulder will read the files on six unsolved cases from a list of thirteen. He does not care about the order in which he reads the files.
The number of ways to perform the task is Please (Do not evaluate.) 28. Specify the number of ways to perform the task described. Give your answers using P(n, r) or C(n, r) notation.
The editorial staff of a literary magazine has received thirteen articles. The staff wants to choose six articles for the next issue of the magazine.
The number of ways to perform the task is . (Do not evaluate.) 29. Of 12 possible books, you plan to take 6 with you on vacation. How many different collections of 6 books can you take?
You can take different collections of 6 books on vacation with you. 30. The academic computing committee at a college is in the process of evaluating different computer systems. The committee consists of seven administrators, eight faculty, and four students. A seven­
person subcommittee is to be formed. The subcommittee must have a chair and vice chair from the administrators, the other five committee members have no particularly defined roles from the faculty and students. In how many ways can this subcommittee be formed?
The subcommittee can be formed in ways.

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