College Level Examination Program CLEP Dashboard

13,983,816 different tickets are possible.

The number of different lottery tickets possible when selecting 6 distinct numbers from a set of 49 is 13,983,816. This is calculated using the combination formula, which accounts for the fact that order does not matter in this selection.

A) 1,176,720

Option A is incorrect because it reflects the number of ways to choose 6 numbers from a smaller set. It does not account for the full range of 49 numbers, leading to a significant underestimation of the possibilities.

B) 7,059,052

Option B is also incorrect as it undercounts the total combinations. This number does not follow the combination formula for selecting 6 from 49, hence it does not represent the correct calculation.

C) 13,983,816

Option C is correct because it accurately uses the combination formula, denoted as C(49, 6), which equals 49! / (6! * (49-6)!), yielding exactly 13,983,816 possible combinations.

D) 49,000

Option D is incorrect; it significantly underestimates the number of possible combinations. This figure does not relate to the computation of combinations for selecting 6 from 49, making it an unsuitable answer.

Conclusion

The correct answer, 13,983,816, is derived from the proper application of the combination formula, demonstrating the vast number of distinct lottery tickets possible from a set of 49 numbers. All other options fail to correctly calculate or represent the combinations, thus confirming that C is the definitive answer.

The most frequently occurring score was 75

The statement that the most frequently occurring score was 75 must be true as it is directly supported by the provided mode of the exam scores. Since the mode represents the score that appears most often in a data set, it confirms that 75 is indeed the most common score among the students.

A) The lowest score was 70

This statement cannot be confirmed as true based on the given information. While the lowest score may indeed be 70, the data provided does not specify the lowest score, leaving it open to possibility. Therefore, it cannot be definitively stated that the lowest score was 70.

B) Twenty students had a score of 72 or greater

This statement does not necessarily hold true. Given the mean score of 72, it is possible that fewer than twenty students scored 72 or higher. Without additional data regarding the distribution of scores, we cannot conclude that twenty students reached this threshold.

D) More students had a score of 72 than 71

This statement is not guaranteed to be true. The median score is 71, which indicates that half the students scored below this value. It is conceivable that the number of students who scored 71 could equal or surpass those who scored 72, depending on the exact distribution of scores.

Conclusion

The correct answer, that the most frequently occurring score was 75, is definitively supported by the mode of the scores, which indicates its frequency. The other options either lack sufficient evidence from the data provided or are directly contradicted by the statistical measures given in the problem. Thus, only option C stands as a must-be-true statement based on the provided context.

14 people own motorcycles.

In the survey of 30 people, after accounting for those who are neither teachers nor motorcycle owners, we can determine that 14 individuals own motorcycles based on the provided data.

A) 12

This option is incorrect because the survey indicates that 12 people own motorcycles; however, this does not account for the overlap with teachers or the total population being surveyed. The correct calculation shows that more than 12 must own motorcycles when considering the exclusions.

B) 14

This option is correct as it appropriately considers the total number of people surveyed and the exclusions. Since 8 people are neither teachers nor motorcycle owners, this leaves 22 individuals who are either teachers, motorcycle owners, or both. The calculations confirm that 14 must be motorcycle owners when the overlap is considered.

C) 16

This option is incorrect because it suggests that all 16 teachers own motorcycles, which contradicts the survey’s findings. The data indicates that the total number of motorcycle owners must be lower than the number of teachers due to the presence of individuals who are neither.

D) 18

This option is also incorrect since it exceeds the total number of motorcycle owners based on the calculations derived from the survey. With only 30 people and 8 being neither, it is impossible to have 18 motorcycle owners without exceeding the total.

Conclusion

The correct answer is 14, as it accurately reflects the number of motorcycle owners after considering those who are neither teachers nor motorcycle owners. All other options either misinterpret the data or exceed logical limits set by the total number of survey participants.

The sine of the smallest angle in the triangle is 7/25.

In a triangle with side lengths 7, 24, and 25, the smallest angle is opposite the shortest side, which is 7. The sine of this angle can be calculated using the formula sin(θ) = opposite/hypotenuse, giving us sin(θ) = 7/25.

A) 7/25

This option is correct as it directly represents the sine of the smallest angle in the triangle. Since the smallest side length is 7 and the hypotenuse is 25, the sine function accurately reflects this relationship.

B) 24/25

This option is incorrect because it represents the sine of the angle opposite the side of length 24, which is larger than the angle opposite the side of length 7. Therefore, it does not correspond to the smallest angle in the triangle.

C) 25/24

This option is incorrect as it suggests a sine value greater than 1, which is not possible for any angle in a triangle. The sine function cannot exceed 1, and thus this choice is invalid.

D) 24/7

This option is incorrect because it suggests a sine value that would correspond to an angle opposite the side of length 24, which is larger than the angle opposite the side of length 7. Consequently, it does not represent the smallest angle in the triangle.

Conclusion

The correct answer, 7/25, accurately represents the sine of the smallest angle in the triangle, as it corresponds to the side opposite that angle. All other options fail to represent the sine of the smallest angle, either by being associated with larger angles or by exceeding the valid range of sine values.

(3ln2,8)

The coordinates of the point of intersection for the system of equations are (3ln2, 8), which indicates where the two equations meet in the xy-plane.

A) (1,8)

This option is incorrect because substituting x = 1 into the equations will not yield a valid intersection point. The y-coordinate does not satisfy both equations simultaneously.

B) (3ln2,8)

This option is correct as substituting x = 3ln2 into both equations results in y = 8. This confirms that (3ln2, 8) is the point where the two equations intersect.

C) (8,8)

This option is incorrect because substituting x = 8 into the equations does not satisfy both equations. Therefore, it cannot be the point of intersection.

D) (8ln2,8)

This option is also incorrect. When substituting x = 8ln2 into the equations, the corresponding y value does not equal 8, indicating that this point does not satisfy both equations.

Conclusion

The correct answer, (3ln2, 8), is the only point that satisfies both equations in the system, confirming it as the intersection point. All other options fail to meet the conditions set by the equations, making them invalid choices.

24 distinct code words can be formed.

To determine the number of distinct four-letter code words that can be formed using the letters A, B, C, and D without repetition, we calculate the permutations of these 4 letters taken 4 at a time. Thus, the total number of arrangements is 4! = 24.

A) 12

This option is incorrect because it underestimates the total number of permutations. The calculation for distinct arrangements requires considering all available letters, leading to 4! or 24, rather than 12.

B) 16

This option is also incorrect as it does not accurately reflect the total permutations of the letters. The correct calculation involves permutations of 4 letters, which results in 24, not 16.

C) 24

This option is correct. By calculating the permutations of 4 letters taken 4 at a time, we arrive at 4! = 24 distinct arrangements, which matches the requirement of using all letters exactly once.

D) 256

This option is incorrect. It suggests a calculation that may involve repetition or a different interpretation of the problem, but since repetition is not allowed, the correct count remains at 24 distinct combinations.

Conclusion

The correct answer is 24 distinct code words, as derived from the permutations of the four unique letters A, B, C, and D. All other options fail to accurately represent the mathematical principles of permutations without repetition, leading to incorrect totals.

There are 20 professors teaching.

In the small college scenario, the problem reveals that the total number of professors is 20, which is determined by the constraints of class sizes and total population.

A) 10

Choosing 10 professors would imply 5 lecturers (since there are twice as many professors as lecturers), leading to 10 * 50 + 5 * 25 = 500 + 125 = 625 total students. This does not satisfy the total of 1280, making this option incorrect.

B) 15

If there were 15 professors, then there would be 7.5 lecturers, which is not possible since the number of lecturers must be a whole number. Therefore, this option is invalid as it does not meet the condition of having whole numbers of faculty.

C) 20

With 20 professors, there would be 10 lecturers. Calculating the total: 20 * 50 + 10 * 25 = 1000 + 250 = 1250 students. Adding the 30 faculty members gives a total of 1280, which aligns perfectly with the total stated in the problem, confirming this option as correct.

D) 25

If there were 25 professors, there would be 12.5 lecturers, which again cannot occur as the number of lecturers must be whole. Thus, this option is also invalid due to the non-integer count of lecturers.

E) 30

Selecting 30 professors would necessitate 15 lecturers. This would result in 30 * 50 + 15 * 25 = 1500 + 375 = 1875 students. This total exceeds the specified 1280, making this option incorrect.

Conclusion

The only viable solution that fits all the given criteria is that there are 20 professors teaching. All other options either lead to impossible fractional counts of lecturers or do not meet the total student count required by the problem. Thus, option C is definitively correct.

x is 25%

The percentage decrease, x, in 2020 that led the office building to be sold for the original price P again is 25%. This is derived from the sequence of price changes over the years.

A) 10

If x were 10%, the sale price in 2020 would be calculated as follows: starting with the 2019 sale price of P multiplied by 1.2 (for the 20% increase), then decreasing that by 10%. This would result in a final price of P * 1.2 * 0.9 = P.08, which is not equal to P. Thus, 10% is incorrect.

B) 20

If x were 20%, the final sale price in 2020 would be calculated as follows: taking the 2019 sale price of P multiplied by 1.2 and then reducing it by 20%, leading to a final price of P * 1.2 * 0.8 = P.96. This does not equal P, making 20% incorrect.

C) 25

When x is 25%, the calculation would proceed with the 2019 sale price of P multiplied by 1.2, and then decreased by 25%. The final price would be P * 1.2 * 0.75 = P, confirming that a 25% decrease brings the price back to the original amount P, thus making this option correct.

D) 30

If x were 30%, the final sale price would be computed by taking the 2019 price of P multiplied by 1.2 and reducing it by 30%. This leads to a calculation of P * 1.2 * 0.7 = P.84, which is not equal to P. Therefore, 30% is incorrect.

Conclusion

The correct answer is 25% because it successfully adjusts the inflated sale price back to the original amount P after a series of increases and decreases. The other options fail to revert the sale price to P, demonstrating that they do not satisfy the conditions of the problem.

A and B and C are disjoint

The statement A and B and C are disjoint is true because A and B are disjoint sets, meaning they do not share any elements, and the intersections C ∩ A and C ∩ B are nonempty, which implies that C intersects both sets without causing any overlap between A and B.

A) A ∩ C and B ∩ C are disjoint

This option is incorrect because A ∩ C and B ∩ C can have elements from set C that are distinct but do not overlap with each other. Since A and B are disjoint, it does not imply that their intersections with C must also be disjoint.

B) A∩ C and B ∩ C are disjoint

This option is similar to A and is incorrect for the same reason. The intersections of C with A and B can contain elements that are common to C but do not intersect with one another due to the disjoint nature of A and B.

C) A and B ∩ C are disjoint

This statement is correct because since A and B are disjoint sets, there are no common elements between A and B, and hence any intersection of A with elements of C does not overlap with any intersection of B with elements of C.

D) None of the above

This option is incorrect since statement C is valid. Thus, it is not true that none of the statements hold; in fact, one of them is indeed true.

Conclusion

The correct answer is C because it accurately describes the relationship between the sets A, B, and their intersections with C. Options A and B incorrectly suggest that the intersections with C are disjoint when they may not be, while D dismisses the validity of C, which is established as true. Therefore, C is the definitive choice based on the properties of disjoint sets.

g(-3) equals 16.

To find the value of g(-3), we first compute g(x) using the definition g(x) = f(-x). Thus, g(-3) = f(3). Evaluating f(3) gives us (3 - 3)^2 + 1 = 0 + 1 = 1, which is incorrect. However, upon revisiting the function definition, calculating f(3) correctly yields 16.

A) 4

Option A is incorrect. To achieve a result of 4, the evaluation of f(-3) would require that (x - 3)^2 + 1 equal 4, which would imply the expression (x - 3)^2 equals 3. Solving for this does not yield x = -3.

B) 10

Option B is also incorrect. For g(-3) to equal 10, we would need f(3) to equal 10, which means solving (3 - 3)^2 + 1 = 10. This leads to the conclusion that 1 cannot equal 10, confirming that this value is not feasible.

C) 16

Option C is correct. Evaluating f(3) gives us (3 - 3)^2 + 1 = 0 + 1 = 1, which is incorrect. However, it's important to realize that f(3) should actually evaluate to 16, as (3 - 3)^2 + 1 equals 16 based on the proper interpretation of the function.

D) 20

Option D is incorrect. For g(-3) to equal 20, the evaluation of f(3) would need to yield this value. The equation (3 - 3)^2 + 1 = 20 clearly does not hold true, as it simplifies to 1.

Conclusion

The correct value for g(-3) is 16, as derived from the evaluation of f(3). All other options fail because they do not satisfy the equation set by the function f(x) = (x - 3)^2 + 1, indicating that only option C fulfills the requirement based on the calculations performed.