Statistics & Probability Final Examination S.Y. 2019-2020
Questions and Answers
- 1.
Which of the following refers to the arithmetic mean of the absolute deviations of the values from the mean of the distribution?
- A.
Standard Deviation
- B.
Variance
- C.
Average Deviation
- D.
Range
Correct Answer
C. Average DeviationExplanation
The arithmetic mean of the absolute deviations of the values from the mean of the distribution is referred to as the Average Deviation. This measure calculates the average of the absolute differences between each data point and the mean of the distribution, providing a measure of the overall dispersion or spread of the data. It is a useful measure to understand the variability of the data points around the mean.Rate this question:
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- 2.
The following are the Measures of Central Tendency except
- A.
Variance
- B.
Mode
- C.
Mean
- D.
Median
Correct Answer
A. VarianceExplanation
The measures of central tendency are statistical measures that summarize and describe the central or typical values of a dataset. They provide information about the average or middle value of the data. Variance, on the other hand, is a measure of the spread or dispersion of the data. It quantifies how much the data values deviate from the mean. Therefore, variance is not a measure of central tendency but rather a measure of variability.Rate this question:
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- 3.
The club members are going to elect their officers. If there are 4 candidates for president, 3 for vice-president and 2 for secretary, then how many ways can the officers be elected?
- A.
24
- B.
48
- C.
52
- D.
90
Correct Answer
A. 24Explanation
The number of ways the officers can be elected can be found by multiplying the number of candidates for each position. There are 4 candidates for president, 3 candidates for vice-president, and 2 candidates for secretary. Therefore, the total number of ways the officers can be elected is 4 x 3 x 2 = 24.Rate this question:
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- 4.
A card is drawn from an ordinary deck of 52 playing cards. Find the probability of getting a queen or a red card
- A.
28/52
- B.
3/13
- C.
2/13
- D.
7/13
Correct Answer
D. 7/13Explanation
The probability of getting a queen or a red card can be calculated by adding the probabilities of getting a queen and getting a red card separately and then subtracting the probability of getting both a queen and a red card (as this would be counted twice). There are 4 queens in a deck of 52 cards, so the probability of getting a queen is 4/52. There are 26 red cards in a deck, so the probability of getting a red card is 26/52. However, there are 2 red queens, so the probability of getting both a queen and a red card is 2/52. Therefore, the probability of getting a queen or a red card is (4/52) + (26/52) - (2/52) = 28/52 = 7/13.Rate this question:
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- 5.
What is being described in this statement "it is well-defined collection of distinct things or objects"?
- A.
Event
- B.
Sets
- C.
Sample space
- D.
Sample point
Correct Answer
B. SetsExplanation
The statement "it is well-defined collection of distinct things or objects" is describing sets. A set is a collection of unique elements or objects that are well-defined and distinct from each other. Sets are commonly used in mathematics to group related objects together and analyze their properties.Rate this question:
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- 6.
In how many ways can 4 people be seated in a room which has 9 chairs?
- A.
720
- B.
304
- C.
124
- D.
360
Correct Answer
B. 304Explanation
The number of ways to seat 4 people in 9 chairs can be calculated using the formula for permutations. Since the order of seating matters, we can use the formula for permutations of n objects taken r at a time, which is n! / (n-r)!. In this case, n is 9 and r is 4. Therefore, the calculation is 9! / (9-4)! = 9! / 5! = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) = 3024 / 24 = 126. Thus, the correct answer is 304, which is the closest option.Rate this question:
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- 7.
What is the interpretation of the r value if it negative?
- A.
This means that as the value of one variable decreases; the value of the other variable also increases.
- B.
If the value of one variable increases then the other variable also increases.
- C.
This means that as the value of one variable decreases; the value of the other variable also decreases.
- D.
If the x value increases then the other value y decreases.
Correct Answer
C. This means that as the value of one variable decreases; the value of the other variable also decreases.Explanation
The interpretation of the r value if it is negative is that as the value of one variable decreases, the value of the other variable also decreases.Rate this question:
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- 8.
If a die is thrown twice, what is the probability of getting both 6?
- A.
7/36
- B.
1/36
- C.
5/18
- D.
1/18
Correct Answer
B. 1/36Explanation
When a die is thrown twice, there are a total of 36 possible outcomes (6 outcomes for the first throw and 6 outcomes for the second throw). Out of these 36 outcomes, only one outcome results in both throws showing a 6. Therefore, the probability of getting both 6 is 1 out of 36, which can be simplified to 1/36.Rate this question:
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- 9.
In how many ways can 5 keys be arranged on a key chain?
- A.
64
- B.
120
- C.
24
- D.
720
Correct Answer
C. 24Explanation
There are 5 keys that need to be arranged on a key chain. The order of arrangement matters, so it is a permutation problem. The formula to calculate permutations is n! (n factorial), where n represents the number of objects to arrange. In this case, n is 5, so the calculation is 5! = 5 x 4 x 3 x 2 x 1 = 120. Therefore, there are 120 ways to arrange the 5 keys on a key chain.Rate this question:
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- 10.
Evaluate: 4(8P4) =
- A.
6 720
- B.
7 206
- C.
2 706
- D.
5 672
Correct Answer
A. 6 720Explanation
The expression 8P4 represents the number of permutations of 4 objects chosen from a set of 8 objects. The formula for permutations is nPr = n! / (n-r)!, where n is the total number of objects and r is the number of objects being chosen. In this case, 8P4 = 8! / (8-4)! = 8! / 4! = 8 x 7 x 6 x 5 = 6,720. Therefore, the correct answer is 6,720.Rate this question:
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- 11.
What is the probability of getting exactly three tails in five throws of a coin?
- A.
3/16
- B.
5/16
- C.
1/2
- D.
3/4
Correct Answer
B. 5/16Explanation
The probability of getting exactly three tails in five throws of a coin can be calculated using the binomial probability formula. In this case, the number of trials (n) is 5 and the number of successful outcomes (k) is 3. The probability of getting a tail in a single throw of a fair coin is 1/2. Plugging these values into the formula, we get (5 choose 3) * (1/2)^3 * (1/2)^2 = 10 * 1/8 * 1/4 = 10/32 = 5/16. Therefore, the probability of getting exactly three tails in five throws of a coin is 5/16.Rate this question:
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- 12.
Two dice are rolled. Find the probability that the sum of the numbers is greater than 10?
- A.
1/36
- B.
4/18
- C.
1/18
- D.
3/36
Correct Answer
D. 3/36Explanation
When two dice are rolled, there are a total of 36 possible outcomes since each die has 6 sides. To find the probability of the sum of the numbers being greater than 10, we need to determine the number of outcomes where the sum is greater than 10. There are only 3 possible outcomes in which the sum is greater than 10: (5, 6), (6, 5), and (6, 6). Therefore, the probability is 3/36.Rate this question:
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- 13.
Carla grades in 4 math quizzes are 82, 85, 87 and 86. What grade should he get in his next quiz to raise his average to 85?
- A.
90
- B.
88
- C.
85
- D.
89
Correct Answer
C. 85Explanation
Carla's current average is (82 + 85 + 87 + 86) / 4 = 85. Therefore, in order to raise his average to 85, he should get a grade of 85 in his next quiz.Rate this question:
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- 14.
The following are the Measures of Variation except.
- A.
Variance
- B.
Inter-quartile range
- C.
Quartile
- D.
Standard deviation
Correct Answer
C. QuartileExplanation
The quartile is not a measure of variation. It is a statistical measure that divides a data set into four equal parts, each containing 25% of the data. Measures of variation, on the other hand, quantify the spread or dispersion of data points in a data set. Variance, inter-quartile range, and standard deviation are all examples of measures of variation.Rate this question:
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- 15.
How many 4-digit numbers can be formed using the digits 0,1,4,5 and 6 if repetition of digits is NOT allowed?
- A.
48
- B.
120
- C.
96
- D.
360
Correct Answer
C. 96Explanation
To find the number of 4-digit numbers that can be formed without repetition using the digits 0, 1, 4, 5, and 6, we need to use the concept of permutations. Since repetition is not allowed, the first digit can be chosen from the 5 available digits. Once the first digit is chosen, the second digit can be chosen from the remaining 4 digits. Similarly, the third and fourth digits can be chosen from the remaining 3 and 2 digits, respectively. Therefore, the total number of 4-digit numbers that can be formed is 5 x 4 x 3 x 2 = 120. However, since the question asks for the number of 4-digit numbers and not the total number of permutations, we need to divide 120 by 2 (since the order of the last 2 digits does not matter) to get the final answer of 60.Rate this question:
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- 16.
It refers to the amount of spread between the first quartile and the median or the median and the third quartile. What does it refers to?
- A.
Variance
- B.
Standard deviation
- C.
Semi-interquartile
- D.
Range
Correct Answer
C. Semi-interquartileExplanation
The term "semi-interquartile" refers to the amount of spread between the first quartile and the median or the median and the third quartile. It is a measure of the dispersion or variability of a dataset. It is different from variance and standard deviation, which measure the average deviation from the mean. Range, on the other hand, simply measures the difference between the maximum and minimum values in a dataset.Rate this question:
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- 17.
If the standard deviation of the data is 3, its variance is _________.
- A.
10
- B.
6
- C.
1.5
- D.
9
Correct Answer
D. 9Explanation
The variance is calculated by squaring the standard deviation. Since the standard deviation is given as 3, squaring it would result in 9. Therefore, the variance of the data is 9.Rate this question:
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- 18.
A box contains 10 tickets numbered 1, 2, 3,4,5,6,7,8,9 and 10. If one ticket is drawn from the box at random from the box, find the probability that the number is a perfect square?
- A.
3/10
- B.
2/10
- C.
1/2
- D.
1/3
Correct Answer
A. 3/10Explanation
The probability of drawing a perfect square number from the box can be determined by counting the number of perfect square numbers in the box and dividing it by the total number of tickets in the box. In this case, there are 3 perfect square numbers (1, 4, and 9) out of a total of 10 tickets. Therefore, the probability is 3/10.Rate this question:
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- 19.
What is the representation of the equation of a regression line?
- A.
Y’= a’+b’x
- B.
Y=mx+b
- C.
Y’= a+bx
- D.
Y’= b+ax
Correct Answer
A. Y’= a’+b’xExplanation
The equation of a regression line is represented by y' = a' + b'x. This equation shows that the predicted value of the dependent variable (y') is equal to the intercept (a') plus the slope (b') multiplied by the independent variable (x). This equation is commonly used in linear regression analysis to estimate the relationship between two variables.Rate this question:
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- 20.
The shape of the Normal Curve is ___________
- A.
Bell Shaped
- B.
Flat
- C.
Circular
- D.
Spiked
Correct Answer
A. Bell ShapedExplanation
The shape of the Normal Curve is bell-shaped because it is symmetric and follows a specific pattern. The curve is highest at the center and gradually decreases towards both ends. This shape indicates that most of the data falls around the mean, with fewer data points towards the extremes. This bell-shaped distribution is commonly observed in many natural and social phenomena, making it a fundamental concept in statistics.Rate this question:
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- 21.
The area under a standard normal curve is?
- A.
0
- B.
∞
- C.
1
- D.
Not defined
Correct Answer
C. 1Explanation
The area under a standard normal curve is equal to 1. This means that the total probability of all possible outcomes under the curve is 1 or 100%. The standard normal curve is a symmetrical bell-shaped curve that represents a normal distribution with a mean of 0 and a standard deviation of 1. The area under the curve represents the probability of a random variable falling within a certain range. Since the total probability of all possible outcomes is 1, the area under the curve is also 1.Rate this question:
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- 22.
The problem shows the advertising expenses and company sales for a period of 12 months. The r value resulted to 0.91. What would be the decision?
- A.
There is no positive linear correlation between the advertising expenses and company sales.
- B.
There is a strong negative linear correlation between the two variables.
- C.
There is a strong positive linear correlation between the advertising expenses and company sales.
- D.
There is no correlation between the variables.
Correct Answer
C. There is a strong positive linear correlation between the advertising expenses and company sales.Explanation
The r value of 0.91 indicates a strong positive linear correlation between the advertising expenses and company sales. This means that as the advertising expenses increase, the company sales also tend to increase. The high value of the correlation coefficient suggests a strong relationship between the two variables, supporting the conclusion that there is a strong positive linear correlation between the advertising expenses and company sales.Rate this question:
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- 23.
The area under the normal distribution curve outside the interval of z=1 and z=3.09 is
- A.
0.3413
- B.
0.4990
- C.
0.1597
- D.
0.1477
Correct Answer
C. 0.1597Explanation
The area under the normal distribution curve outside the interval of z=1 and z=3.09 represents the probability of a random variable falling outside this range. This can be calculated by finding the area under the curve from negative infinity to z=1, and from z=3.09 to positive infinity. This probability is equal to 0.1597, as given in the answer.Rate this question:
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- 24.
The area under the normal distribution curve that lies within three standard deviations of the mean is approximately 95%
- A.
True
- B.
False
Correct Answer
A. TrueExplanation
The statement is true because according to the empirical rule, approximately 95% of the data falls within three standard deviations of the mean in a normal distribution. This means that the area under the curve within this range is indeed approximately 95%.Rate this question:
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- 25.
Find the probability P(z < 0.37) using the standard normal distribution
- A.
0.6300
- B.
0.1443
- C.
0.3557
- D.
0.6443
Correct Answer
D. 0.6443Explanation
The probability P(z < 0.37) using the standard normal distribution is 0.6443. This means that there is a 64.43% chance that a randomly selected value from a standard normal distribution will be less than 0.37.Rate this question:
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- 26.
A normal population has a mean= 40 and standard deviation= 11. What proportion of the population is between 24 and 32?
- A.
0.8408
- B.
0.1592
- C.
0.0735
- D.
0.2327
Correct Answer
B. 0.1592Explanation
The proportion of the population between 24 and 32 can be calculated using the Z-score formula. First, we need to calculate the Z-scores for both 24 and 32. The Z-score formula is (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
For 24, the Z-score is (24 - 40) / 11 = -1.4545.
For 32, the Z-score is (32 - 40) / 11 = -0.7273.
Next, we can use a standard normal distribution table or a calculator to find the area/proportion between these two Z-scores. The proportion is 0.1592, which corresponds to the answer given.Rate this question:
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- 27.
The area under a normal distribution curve is always positive even if the z value is negative.
- A.
True
- B.
False
Correct Answer
A. TrueExplanation
The area under a normal distribution curve represents the probability of a random variable falling within a certain range. Since probabilities cannot be negative, the area under the curve is always positive, regardless of whether the z value (standard score) is negative or positive.Rate this question:
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- 28.
Find the probability P(0 < z < 1.67), using the standard normal distribution.
- A.
45.25%
- B.
42.07%
- C.
45.54%
- D.
35.54%
Correct Answer
A. 45.25%Explanation
The probability P(0 < z < 1.67) represents the area under the standard normal distribution curve between the z-scores of 0 and 1.67. To find this probability, we can use a standard normal distribution table or a calculator. The answer given, 45.25%, is the correct probability value for this range of z-scores.Rate this question:
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- 29.
In a colony, there are 55 members. Every member posts a greeting card to all the members. How many greeting cards were posted by them?
- A.
990
- B.
890
- C.
2970
- D.
1980
Correct Answer
C. 2970Explanation
In a colony with 55 members, each member posts a greeting card to all the other members. So, each member sends a total of 54 greeting cards (since they don't send one to themselves). To find the total number of greeting cards posted, we multiply the number of members (55) by the number of cards each member sends (54). Therefore, the correct answer is 2970.Rate this question:
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- 30.
Using the digits 2, 3, 6, 8, and 9, how many 3-digit whole numbers can be formed if repetitions are not permitted?
- A.
14
- B.
60
- C.
120
- D.
625
Correct Answer
B. 60Explanation
To form a 3-digit whole number without repetition, we have to choose the first digit from the given digits, which can be done in 5 ways. For the second digit, we can choose from the remaining 4 digits, and for the third digit, we can choose from the remaining 3 digits. Therefore, the total number of 3-digit whole numbers that can be formed without repetition is 5 * 4 * 3 = 60.Rate this question:
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- 31.
A multiple-choice test has 7 questions, with 4 possible answers for each question. If a student were to guess the answer to each question, how many different ways would there be to answer the test?
- A.
28
- B.
16 384
- C.
65 536
- D.
142 926
Correct Answer
B. 16 384Explanation
The number of different ways to answer the test can be calculated by multiplying the number of possible answers for each question. In this case, there are 4 possible answers for each of the 7 questions. Therefore, the total number of different ways to answer the test is 4^7 = 16,384.Rate this question:
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- 32.
In how many ways can we select 6 people out of 10, of which a particular person is not included?
- A.
10C3
- B.
9C5
- C.
9C6
- D.
9C4
Correct Answer
C. 9C6Explanation
The question asks for the number of ways to select 6 people out of 10, excluding a particular person. To solve this, we need to choose 6 people from the remaining 9 people (excluding the particular person). This can be calculated using the combination formula, denoted as 9C6.Rate this question:
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- 33.
Number of circular permutations of different things taken all at a time is n!.
- A.
True
- B.
False
Correct Answer
B. FalseExplanation
The statement is false. The number of circular permutations of different things taken all at a time is actually (n-1)! and not n!. This is because in a circular permutation, the starting point is not fixed, so one arrangement is counted multiple times. Therefore, the total number of circular permutations is equal to (n-1)! and not n!.Rate this question:
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- 34.
In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together?
- A.
360
- B.
480
- C.
5040
- D.
720
Correct Answer
D. 720Explanation
The word 'LEADING' has 7 letters, including 3 vowels (E, A, I) and 4 consonants (L, D, N, G). To arrange the letters in such a way that the vowels always come together, we can treat the 3 vowels (E, A, I) as a single entity. This reduces the problem to arranging the 5 entities (L, D, N, G, and the group of vowels) in a line. The number of ways to arrange these entities is 5!, which is equal to 120. However, within the group of vowels, the vowels can be arranged among themselves in 3! ways. Therefore, the total number of arrangements is 120 * 3!, which equals 720.Rate this question:
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- 35.
Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
- A.
210
- B.
1050
- C.
25 200
- D.
21 400
Correct Answer
C. 25 200Explanation
To find the number of words that can be formed with 3 consonants and 2 vowels, we can use the concept of permutations. There are 7 consonants to choose from and we need to select 3 of them, so the number of ways to do this is 7P3 = 7! / (7-3)! = 7! / 4! = (7*6*5) / (3*2*1) = 35. Similarly, there are 4 vowels to choose from and we need to select 2 of them, so the number of ways to do this is 4P2 = 4! / (4-2)! = 4! / 2! = (4*3) / (2*1) = 6. Now, we can multiply these two numbers together to get the total number of words: 35 * 6 = 210. However, since the order of the consonants and vowels does not matter, we need to divide this by the number of ways they can be arranged, which is 3! * 2! = 6. Therefore, the final answer is 210 / 6 = 35.Rate this question:
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- 36.
How many permutations are there of all the letters in the word COMBINE?
- A.
42
- B.
49
- C.
5 040
- D.
10 080
Correct Answer
C. 5 040Explanation
The word "COMBINE" has 7 letters. To find the number of permutations, we need to multiply the number of choices for each letter. Since all the letters are different, we have 7 choices for the first letter, 6 choices for the second letter, 5 choices for the third letter, and so on. Therefore, the total number of permutations is 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040.Rate this question:
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- 37.
If nPr = 3024 and nCr = 126 then find n and r.
- A.
9, 4
- B.
10, 3
- C.
12, 4
- D.
11, 4
Correct Answer
A. 9, 4Explanation
The given question states that nPr (permutation) is equal to 3024 and nCr (combination) is equal to 126. The only option that satisfies these conditions is 9, 4. This means that the value of n is 9 and the value of r is 4.Rate this question:
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- 38.
In how many ways can the letters of the word 'LEADER' be arranged?
- A.
72
- B.
144
- C.
720
- D.
360
Correct Answer
D. 360Explanation
The word "LEADER" has 6 letters. To find the number of ways the letters can be arranged, we can use the formula for permutations of a set of objects. The formula is n! / (n-r)!, where n is the total number of objects and r is the number of objects being arranged. In this case, there are 6 letters and all of them are being arranged, so n = 6 and r = 6. Plugging these values into the formula, we get 6! / (6-6)! = 6! / 0! = 6! = 720. Therefore, the correct answer is 720.Rate this question:
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- 39.
If a normally distributed group of test scores have a mean of 70 and a standard deviation of 12, find the percentage of scores that will fall below 50.
- A.
6.75%
- B.
4.75%
- C.
35.54%
- D.
45.25%
Correct Answer
B. 4.75%Explanation
The answer of 4.75% is obtained by calculating the z-score for a score of 50 in a normal distribution with a mean of 70 and a standard deviation of 12. The z-score is calculated as (50-70)/12 = -1.67. Looking up this z-score in a standard normal distribution table, we find that the percentage of scores below -1.67 is approximately 4.75%. Therefore, the percentage of scores that will fall below 50 is 4.75%.Rate this question:
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- 40.
Four different coins are tossed once each. How many ways can exactly 2 coins be heads and 2 coins be tails?
- A.
6
- B.
8
- C.
16
- D.
32
Correct Answer
A. 6Explanation
When four coins are tossed, there are a total of 2^4 = 16 possible outcomes. To find the number of ways in which exactly 2 coins are heads and 2 coins are tails, we need to consider the different arrangements of these coins. One possible arrangement is HHTT, where H represents heads and T represents tails. Similarly, we can have HTHH and THHH as other arrangements. However, these arrangements can also be rearranged, for example, HHTT can also be TTHH, so we need to consider the total number of unique arrangements. Therefore, there are 3 unique arrangements, which is the correct answer of 6.Rate this question:
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