Question 1

Solve the problem.
-A committee is to be formed consisting of 5 men and 2 women. If the committee members are to be chosenfrom 13 men and 9 women, how many different committees are possible?
A)46,332
B)170,544
C)11,119,680
D)1,323

Accepted Answer

To form the committee, choose 5 men from 13 and 2 women from 9. The number of ways to choose 5 men from 13 is C(13,5) = 1,287, and the number of ways to choose 2 women from 9 is C(9,2) = 36. Multiply these results to find the total number of different committees: 1,287 * 36 = 46,332.

Question 2

Solve the problem.
-A hot dog stand sells hot dogs with cheese, relish, chili, tomato, onion, mustard, or ketchup. How manydifferent hot dogs can be concocted using any 5 of the extras?
A)21
B)2,520
C)42
D)1,260

Accepted Answer

The answer of Solve the problem.
-A hot dog stand sells...

Question 3

Write the word or phrase that best completes each statement or answers the question.
-How many different 11-letter words (real or imaginary)can be formed from the letters of the wordMISSISSIPPI? Leave your answer in factorial form.

Accepted Answer

The answer of Write the word or phrase that best...

Question 4

Solve the problem.
-In how many ways can 5 people each have different birth months?
A)95,040
B)792
C)248,832
D)60

Accepted Answer

This problem can be solved using the permutation formula since we are arranging 5 unique items (people) out of 12 (months). The formula for permutations is P(n, r) = n! / (n-r)!, where n is the total number of items, and r is the number of items to arrange. Here, n = 12 (months) and r = 5 (people), so P(12, 5) = 12! / (12-5)! = 12! / 7! = 12 x 11 x 10 x 9 x 8 = 95,040.

Question 5

Solve the problem.
-How many 5-card poker hands consisting of three 7ʹs and two cards that are not 7ʹs are possible in a52-card deck?
A)4512
B)2256
C)5304
D)2652

Accepted Answer

There are 4 ways to choose the two non-7 cards from the 45 non-7 cards. For each of these choices, there are 13C3 ways to choose the three 7's. So, the total number of hands is 4 * 13C3 = 4 * 286 = 1144.

Question 6

Solve the problem.
-An exam consists of 9 multiple-choice questions and 6 essay questions. If the student must answer 6 of themultiple-choice questions and 2 of the essay questions, in how many ways can the questions be chosen?
A)1260
B)648
C)261,273,600
D)1,814,400

Accepted Answer

The number of ways to choose 6 multiple-choice questions out of 9 is C(9,6) = 84. 
The number of ways to choose 2 essay questions out of 6 is C(6,2) = 15. 
Therefore, the total number of ways to choose the questions is 84*15 = 1260.

Question 7

Solve the problem.
-Mary finds 8 fish at a pet store that she would like to buy, but she can afford only 5 of them. In how manyways can she make her selection? How many ways can she make her selection if he decides that one of thefish is a must?
A)56; 35
B)6,720; 840
C)336; 210
D)3,360; 420

Accepted Answer

The number of ways Mary can select 5 fish out of 8 is calculated using combinations: 8C5 = 8! / (5!(8-5)!) = 56. If one fish is a must, then she needs to choose 4 out of the remaining 7, which is 7C4 = 35.

Question 8

Solve the problem.
-How many different license plates can be made using 2 letters followed by 3 digits selected from the digits0 through 9, if digits may be repeated but letters may not be repeated?
A)650,000
B)128.968254
C)486,720
D)676,000

Accepted Answer

The answer of Solve the problem.
-How many different license plates...

Question 9

Write the word or phrase that best completes each statement or answers the question.
-An environmental organization has 27 members. Each member will be placed on exactly 1 of 4 teams.Each team will work on a different issue. The first team has 10 members, the second has 5, the third has 4,and the fourth has 8. In how many ways can these teams be formed?
A) 
B) 
C) 
D)

Accepted Answer

The answer of Write the word or phrase that best...

Question 10

Solve the problem.
-List all the combinations of 6 objects a, b, c, d, e, and f taken 2 at a time. What is C(6, 2)?
A)ab, ac, ad, ae, af, bc, bd, be, bf, cd, ce, cf, de, df, ef C(6, 2)= 15
B)ab, ac, ad, ae, af, ba, bc, bd, be, bf, ca, cb, cd, ce, cf, da, db, dc, de, df, ea, eb, ec, ed, ef, fa, fb, fc, fd, fe C(6, 2)= 30
C)aa, ab, ac, ad, ae, af, ba, bb, bc, bd, be, bf, ca, cb, cc, cd, ce, cf, da, db, dc, dd, de, df, ea, eb, ec, ed, ee, ef, fb, fc, fd, fe, ff
C(6, 2)= 36
D)ab, ac, ad, ae, bc, bd, be, cd, ce, cf, de, df C(6, 2)= 12

Accepted Answer

The problem asks for combinations taken 2 at a time, which means we don't count combinations like "aa" or "ba" where the objects are repeated or the order is different. We use the formula for combinations, C(n, r) = n!/(r!(n-r)!) where n is the total number of objects and r is the number of objects we are choosing. So in this case, C(6,2) = 6!/[(2!)(6-2)!] = 15. Looking at the choices, only option A has 15 combinations listed. C(6, 2) represents the number of ways of choosing 2 objects from a set of 6 objects without repetition and order. The formula for C(n, r) is n! / (r!*(n-r)!), where n is the total number of objects and r is the number of objects to choose. From the given options, choice A is the one that lists all possible combinations of 2 objects taken from a set of 6 objects. This list contains 15 items, which matches the value of C(6, 2), so the answer is A.

Question 11

Solve the problem.
-How many different license plates can be made using 3 letters followed by 2 digits selected from the digits0 through 9, if neither letters nor digits may be repeated?
A)1,404,000
B)1,757,600
C)117,000
D)1,123,200

Accepted Answer

There are 26 choices for the first letter, 25 for the second, and 24 for the third. For the digits, there are 10 choices for the first digit and 9 for the second. Multiply these together: 26 * 25 * 24 * 10 * 9 = 1,404,000.

Question 12

Solve the problem.
-From 8 names on a ballot, a committee of 5 will be elected to attend a political national convention. Howmany different committees are possible?
A)56
B)6,720
C)336
D)3,360

Accepted Answer

The number of different committees can be calculated using combinations, as the order of selection does not matter. The formula for combinations is C(n, r) = n! / (r!(n-r)!), where n is the total number of items to choose from, and r is the number of items to choose. Here, n = 8 and r = 5, so C(8, 5) = 8! / (5!3!) = 56.

Question 13

Solve the problem.
-How many different 10-letter words (real or imaginary)can be formed from the letters in the wordPHILOSOPHY?
A)453,600
B)3,628,800
C)907,200
D)45,360

Accepted Answer

There are 10 letters in the word PHILOSOPHY. So, the number of different 10-letter words that can be formed from the letters in the word PHILOSOPHY is 10! = 3,628,800.

Question 14

Solve the problem.
-List all the combinations of 4 objects 1, 2, 3, and 4 taken 3 at a time. What is C(4, 3)?
A)123, 124, 134, 234 C(4, 3)= 4
B)123, 124, 132, 134, 142, 143, 213, 214, 231, 234, 241, 243, 312, 314, 321, 324, 341, 342, 412, 413, 421, 423, 431 C(4, 3)= 24
C)123, 124, 134, 234, 321, 432 C(4, 3)= 6
D)111, 112, 113, 114, 121, 122, 123, 124, 131, 132, 133, 134, 141, 142, 143, 144, 211, 212, 213, 214, 221, 222, 223, 224, 231, 232, 233, 234, 241, 242, 243, 244, 311, 312, 313, 314, 321, 322, 323, 324, 331, 332, 333, 334,
341, 342, 343, 344, 411, 412, 413, 414, 421, 422, 423, 424, 431, 432, 433, 434, 441, 442, 443, 444
C(4, 3)= 64

Accepted Answer

The combinations of 4 objects taken 3 at a time are 123, 124, 134, and 234, which totals 4 combinations, matching the formula for combinations C(n, k) = n! / [k!(n-k)!], where n=4 and k=3, giving C(4, 3) = 4.

Question 15

Solve the problem.
-How many 2-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0? No digit can beused more than once.
A)90
B)1,814,400
C)45
D)3,628,800

Accepted Answer

For a 2-digit number, the first digit can be any of the 9 options (1-9, since 0 cannot be the first digit), and the second digit can be any of the remaining 9 digits (including 0 this time). Thus, the total number of 2-digit numbers is 9 * 9 = 81.

Question 16

Find the value of the combination.
-C(11, 8)
A)165
B)990
C)3,326,400
D)12

Accepted Answer

The combination formula is C(n, k) = n! / [k!(n-k)!]. For C(11, 8), it's 11! / [8!(11-8)!] = 11! / (8!3!) = (11*10*9) / (3*2*1) = 165.

Question 17

Write the word or phrase that best completes each statement or answers the question.
-How many different vertical arrangements are there of 9 flags if 4 are white, 3 are blue, and 2 are red?
A)1260
B)24
C)26
D)126

Accepted Answer

To find the number of arrangements, we can use the formula for permutations with repetition:
n!/n1!n2!...nk!
where n is the total number of objects and n1, n2, ..., nk are the number of objects of each distinct type. In this case, we have:
n = 9, n1 = 4 (white flags), n2 = 3 (blue flags), and n3 = 2 (red flags)
Plugging these values into the formula, we get:
9!/4!3!2! = 1260
Therefore, the answer is choice A, 1260. This is a permutation problem, where order matters. Using the formula for permutations of n objects taken r at a time, we get:

P(9,4) * P(5,3) * P(2,2)

= (9!/(9-4)!) * (5!/(5-3)!) * (2!/(2-2)!)

= 1260. Therefore, there are 1260 different vertical arrangements possible.

Question 18

Find the value of the combination.
-C(7, 7)
A)1
B)5,040
C)1,260
D)0.5

Accepted Answer

C(7, 7) represents choosing 7 items from a set of 7 items, where order does not matter and repetition is not allowed. There is only one way to do this, which is to choose all 7 items. Therefore, the value of C(7, 7) is 1. C(7,7) = 1, which means there is only one possible combination when choosing all 7 items out of a set of 7.

Question 19

Solve the problem.
-How many ways are there to choose a soccer team consisting of 3 forwards, 4 midfield players, and 3defensive players, if the players are chosen from 10 forwards, 7 midfield players, and 5 defensive players?
A)42,000
B)646,646
C)36,288,000
D)165

Accepted Answer

The number of ways to choose the team is calculated by multiplying the combinations for each position: C(10,3) * C(7,4) * C(5,3) = 120 * 35 * 10 = 42,000.

Question 20

Solve the problem.
-A group of 6 friends goes bowling. How many different possibilities are there for the order in which theyplay if the youngest person is to bowl first?
A)120
B)720
C)5
D)6

Accepted Answer

Since the youngest person is fixed to bowl first, we only need to arrange the remaining 5 friends. This is a permutation problem of 5 people, which is calculated as 5! (5 factorial), equal to 5 x 4 x 3 x 2 x 1 = 120.

Quiz 10: Counting and Probability

Solve the problem. -A committee is to be formed consisting of 5 men and 2 women. If the committee members are to be chosenfrom 13 men and 9 women, how many different committees are possible?

Solve the problem. -A hot dog stand sells hot dogs with cheese, relish, chili, tomato, onion, mustard, or ketchup. How manydifferent hot dogs can be concocted using any 5 of the extras?

Write the word or phrase that best completes each statement or answers the question. -How many different 11-letter words (real or imaginary)can be formed from the letters of the wordMISSISSIPPI? Leave your answer in factorial form.

Solve the problem. -In how many ways can 5 people each have different birth months?

Solve the problem. -How many 5-card poker hands consisting of three 7ʹs and two cards that are not 7ʹs are possible in a52-card deck?

Solve the problem. -An exam consists of 9 multiple-choice questions and 6 essay questions. If the student must answer 6 of themultiple-choice questions and 2 of the essay questions, in how many ways can the questions be chosen?

Solve the problem. -Mary finds 8 fish at a pet store that she would like to buy, but she can afford only 5 of them. In how manyways can she make her selection? How many ways can she make her selection if he decides that one of thefish is a must?

Solve the problem. -How many different license plates can be made using 2 letters followed by 3 digits selected from the digits0 through 9, if digits may be repeated but letters may not be repeated?

Solve the problem. -List all the combinations of 6 objects a, b, c, d, e, and f taken 2 at a time. What is C(6, 2) ?

Solve the problem. -How many different license plates can be made using 3 letters followed by 2 digits selected from the digits0 through 9, if neither letters nor digits may be repeated?

Solve the problem. -From 8 names on a ballot, a committee of 5 will be elected to attend a political national convention. Howmany different committees are possible?

Solve the problem. -How many different 10-letter words (real or imaginary) can be formed from the letters in the wordPHILOSOPHY?

Solve the problem. -List all the combinations of 4 objects 1, 2, 3, and 4 taken 3 at a time. What is C(4, 3) ?

Solve the problem. -How many 2-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0? No digit can beused more than once.

Find the value of the combination. -C(11, 8)

Write the word or phrase that best completes each statement or answers the question. -How many different vertical arrangements are there of 9 flags if 4 are white, 3 are blue, and 2 are red?

Find the value of the combination. -C(7, 7)

Solve the problem. -How many ways are there to choose a soccer team consisting of 3 forwards, 4 midfield players, and 3defensive players, if the players are chosen from 10 forwards, 7 midfield players, and 5 defensive players?

Solve the problem. -A group of 6 friends goes bowling. How many different possibilities are there for the order in which theyplay if the youngest person is to bowl first?