Question 1

Write the complete binomial expansion.
-
A) 
B) 
C) 
D)

Accepted Answer

The answer of Write the complete binomial expansion.
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A) 
B) 
C)...

Question 2

Solve the problem using the idea of labeling.
-In how many ways can 11 volunteers be assigned to work in the governor's reelection campaign with 3 assigned to stuffing envelopes, 2 assigned to making phone calls, and 6 assigned to making signs?
A) 3,326,400
B) 1,108,800
C) 3,628,800
D) 4620

Accepted Answer

This problem can be solved using the concept of combinations and permutations. Since the order of selection does not matter within each group but matters across the groups due to different tasks, we use combinations to select the groups and then multiply the results. First, choose 3 volunteers for stuffing envelopes from 11, which can be done in $ \binom{11}{3} $ ways. Then, from the remaining 8, choose 2 for making phone calls, which can be done in $ \binom{8}{2} $ ways. Finally, all 6 remaining will be making signs, but since this is the last group, there's only $ \binom{6}{6} = 1 $ way to choose them. The total number of ways to assign the volunteers is the product of these combinations:$$ \binom{11}{3} \times \binom{8}{2} \times \binom{6}{6} = \frac{11!}{3!8!} \times \frac{8!}{2!6!} \times 1 = \frac{11!}{3!2!6!} = 4620. $$This calculation simplifies to 4620 ways to assign the volunteers to the tasks.

Question 3

Solve the problem.
-In an experiment on the sugar content of fruit, a researcher gathers a sample of 6 apples, 2 bananas, and 5 peaches. In how many ways can 3 pieces of fruit be randomly selected from this sample?
A) 3
B) 1716
C) 286
D) 39

Accepted Answer

The total number of ways to select 3 pieces of fruit from the 13 pieces (6 apples + 2 bananas + 5 peaches) is calculated using the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of items, $k$ is the number of items to choose, and $n!$ denotes the factorial of $n$. Thus, the calculation is $C(13, 3) = \frac{13!}{3!(13-3)!} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286$.

Question 4

Solve the problem.
-A bag contains 7 apples and 5 oranges. If you select 6 pieces of fruit without looking, how many ways can you get 6 apples?
A) 14
B) 35
C) 12
D) 7

Accepted Answer

There are only 7 apples in the bag, so you can choose 6 apples in C(7,6) = 7 ways.

Question 5

Solve the problem.
-How many ways can a class president, a class secretary, and a class treasurer be chosen from a group of 6 candidates?
A) 18
B) 15
C) 120
D) 360

Accepted Answer

The number of ways to choose a class president, secretary, and treasurer from 6 candidates is a permutation of 6 items taken 3 at a time, calculated as 6P3 = 6! / (6-3)! = 6*5*4 = 120.

Question 6

Solve the problem.
-A bag contains 8 apples and 6 oranges. If you select 7 pieces of fruit without looking, how many ways can you get exactly 6 apples?
A) 336
B) 2016
C) 168
D) 56

Accepted Answer

To find the number of ways to get exactly 6 apples out of 7 pieces of fruit selected, we use combinations. First, choose 6 apples out of 8, which is 8C6. Then, choose 1 orange out of 6, which is 6C1. Multiply these two combinations: 8C6 * 6C1 = 28 * 6 = 168.

Question 7

Solve the problem.
-In how many ways can a class president and a class secretary be chosen from a group of 10 candidates?
A) 720
B) 90
C) 20
D) 19

Accepted Answer

The number of ways to choose a class president and a class secretary from 10 candidates is calculated using permutations since the order in which they are chosen matters. The formula for permutations is nPr = n! / (n-r)!, where n is the total number of items, and r is the number of items to choose. Here, n=10 (total candidates) and r=2 (positions to fill), so the calculation is 10P2 = 10! / (10-2)! = 10! / 8! = 10 x 9 = 90.

Question 8

Solve the problem using the idea of labeling.
-How many permutations are there of the letters in the word TENNESSEE?
A) 7560
B) 3780
C) 362,880
D) 81

Accepted Answer

The answer of Solve the problem using the idea of...

Question 9

Write the complete binomial expansion.
-
A) 
B) 
C) 
D)

Accepted Answer

The answer of Write the complete binomial expansion.
-
A) 
B) 
C)...

Question 10

Solve the problem.
-Using the numerals 0 through 9, how many 6-digit serial numbers can be made if the first digit must be 0 or 1 and no numerals may be repeated?
A) 8
B) 13,440
C) 30,240
D) 6048

Accepted Answer

The first digit has 2 choices (0 or 1). The remaining 5 digits are chosen from the remaining 9 digits (since no repetition is allowed), so the number of ways is 2 * 9 * 8 * 7 * 6 * 5 = 30,240.

Question 11

Solve the problem.
-In how many ways can a group of 9 students be selected from 10 students?
A) 10
B) 1
C) 9
D) 90

Accepted Answer

Selecting 9 students out of 10 is equivalent to choosing which 1 student to leave out, which can be done in 10 ways.

Question 12

Write the complete binomial expansion.
-
A) 
B) 
C) 
D)

Accepted Answer

The answer of Write the complete binomial expansion.
-
A) 
B) 
C)...

Question 13

Solve the problem.
-How many different three-digit numbers can be written using digits from the set {1, 2, 3, 4, 5} without any repeating digits?
A) 60
B) 20
C) 120
D) 10

Accepted Answer

For a three-digit number without repeating digits, there are 5 choices for the first digit, 4 remaining choices for the second digit, and 3 remaining choices for the third digit. Multiplying these together gives 5 * 4 * 3 = 60 different numbers.

Question 14

Solve the problem using the idea of labeling.
-In how many ways can 9 officers be assigned with 4 assigned to parking, 3 assigned to security, and 2 assigned to emergencies?
A) 1260
B) 2520
C) 40,320
D) 15,120

Accepted Answer

The number of ways to assign 9 officers to 4 parking, 3 security, and 2 emergencies is 1260. This can be calculated using the formula for combinations: C(n, r) = n! / (r! * (n-r)!), where n is the total number of items and r is the number of items to be selected. In this case, n = 9 and r = 4, so C(9, 4) = 9! / (4! * 5!) = 1260.

Question 15

Solve the problem using the idea of labeling.
-How many permutations are there of the letters in the word BASEBALL?
A) 10,080
B) 40,320
C) 20,160
D) 5040

Accepted Answer

The answer of Solve the problem using the idea of...

Question 16

Solve the problem.
-Using the numerals 0 through 9, how many 5-digit serial numbers can be made if the first digit must be 0 or 1 and numerals may be repeated?
A) 20,000
B) 13,122
C) 80
D) 100,000

Accepted Answer

For the first digit, there are 2 choices (0 or 1). For each of the remaining four positions, there are 10 choices (0 through 9), allowing repetition. Thus, the total number of serial numbers is 2 * 10 * 10 * 10 * 10 = 20,000.

Question 17

Solve the problem.
-In how many ways can 5 of 11 cars be parked in 5 spaces?
A) 55,440
B) 11
C) 462
D) 332,640

Accepted Answer

This is a permutation problem because the order in which the cars are parked matters. The formula for permutations is P(n, r) = n! / (n-r)!, where n is the total number of items to choose from, r is the number of items to choose, and "!" denotes factorial, which means the product of all positive integers up to that number. Here, n = 11 (total cars) and r = 5 (spaces available). So, P(11, 5) = 11! / (11-5)! = 11! / 6! = (11×10×9×8×7×6!) / 6! = 11×10×9×8×7 = 55,440.

Question 18

Solve the problem using the idea of labeling.
-How many permutations are there of the letters in the word GIGGLE?
A) 4320
B) 720
C) 36
D) 120

Accepted Answer

The word GIGGLE has 6 letters, but the letters G and I are repeated. To find the number of unique permutations, we use the formula for permutations of a multiset: $ \frac{n!}{n_1!n_2!...n_k!} $, where $ n $ is the total number of items, and $ n_1, n_2, ..., n_k $ are the counts of each of the repeated items. Here, $ n = 6 $, $ n_1 = 3 $ for G, and $ n_2 = 2 $ for I. So, the number of permutations is $ \frac{6!}{3!2!} = 120 $.

Question 19

Solve the problem.
-How many distinct triangles are determined by 7 points lying on a circle, where the vertices of each triangle are chosen from the 7 points ?
A) 210
B) 35
C) 21
D) 840

Accepted Answer

To determine a triangle, we need to choose 3 points out of the 7. The number of ways to do this is given by the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of points and $k$ is the number of points to choose. Here, $n=7$ and $k=3$, so the calculation is $C(7, 3) = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$.

Question 20

Solve the problem.
-Using the numerals 0 through 9, how many 6-digit serial numbers can end with the numeral 0 if the first digit must be 0 or 1 and no numerals may be repeated?
A) 70
B) 5040
C) 1680
D) 8

Accepted Answer

The first digit can be chosen in 2 ways (either 0 or 1). The last digit is fixed as 0. This leaves 4 positions to fill with any of the remaining 8 digits (since 0 and the chosen first digit are not available, and digits cannot be repeated). Thus, the number of ways to fill these positions is 8 * 7 * 6 * 5 = 1680.

Quiz 11: Sequences, Series, and Probability

Write the complete binomial expansion. -

Solve the problem using the idea of labeling. -In how many ways can 11 volunteers be assigned to work in the governor's reelection campaign with 3 assigned to stuffing envelopes, 2 assigned to making phone calls, and 6 assigned to making signs?

Solve the problem. -In an experiment on the sugar content of fruit, a researcher gathers a sample of 6 apples, 2 bananas, and 5 peaches. In how many ways can 3 pieces of fruit be randomly selected from this sample?

Solve the problem. -A bag contains 7 apples and 5 oranges. If you select 6 pieces of fruit without looking, how many ways can you get 6 apples?

Solve the problem. -How many ways can a class president, a class secretary, and a class treasurer be chosen from a group of 6 candidates?

Solve the problem. -A bag contains 8 apples and 6 oranges. If you select 7 pieces of fruit without looking, how many ways can you get exactly 6 apples?

Solve the problem. -In how many ways can a class president and a class secretary be chosen from a group of 10 candidates?

Solve the problem using the idea of labeling. -How many permutations are there of the letters in the word TENNESSEE?

Write the complete binomial expansion. -

Solve the problem. -Using the numerals 0 through 9, how many 6-digit serial numbers can be made if the first digit must be 0 or 1 and no numerals may be repeated?

Solve the problem. -In how many ways can a group of 9 students be selected from 10 students?

Write the complete binomial expansion. -

Solve the problem. -How many different three-digit numbers can be written using digits from the set {1, 2, 3, 4, 5} without any repeating digits?

Solve the problem using the idea of labeling. -In how many ways can 9 officers be assigned with 4 assigned to parking, 3 assigned to security, and 2 assigned to emergencies?

Solve the problem using the idea of labeling. -How many permutations are there of the letters in the word BASEBALL?

Solve the problem. -Using the numerals 0 through 9, how many 5-digit serial numbers can be made if the first digit must be 0 or 1 and numerals may be repeated?

Solve the problem. -In how many ways can 5 of 11 cars be parked in 5 spaces?

Solve the problem using the idea of labeling. -How many permutations are there of the letters in the word GIGGLE?

Solve the problem. -How many distinct triangles are determined by 7 points lying on a circle, where the vertices of each triangle are chosen from the 7 points ?

Solve the problem. -Using the numerals 0 through 9, how many 6-digit serial numbers can end with the numeral 0 if the first digit must be 0 or 1 and no numerals may be repeated?