Q. In a geometric series, if the first term is 4 and the common ratio is 3, what is the 4th term?
Solution
The nth term of a geometric series is given by a_n = ar^(n-1). Thus, a_4 = 4 * 3^(4-1) = 4 * 27 = 108.
Correct Answer: A — 108
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Q. In a sequence defined by a_n = 2^n, what is the 5th term?
Solution
a_5 = 2^5 = 32.
Correct Answer: A — 16
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Q. In a sequence defined by a_n = 2^n, what is the value of a_5?
Solution
a_5 = 2^5 = 32.
Correct Answer: A — 32
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Q. In an arithmetic series, if the first term is 5 and the last term is 45, and there are 10 terms, what is the common difference?
Solution
Using the formula for the nth term: a_n = a + (n-1)d, we have 45 = 5 + 9d. Solving gives d = 4.
Correct Answer: A — 4
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Q. In how many ways can 3 boys and 2 girls be seated in a row?
Solution
The total number of arrangements is (3+2)! = 5! = 120.
Correct Answer: B — 60
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Q. In how many ways can 3 different colored balls be arranged in a line?
Solution
The number of arrangements of 3 different colored balls is 3! = 6.
Correct Answer: A — 6
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Q. In how many ways can 3 different colored balls be chosen from a set of 7?
Solution
The number of ways to choose 3 from 7 is given by C(7,3) = 7!/(3!4!) = 35.
Correct Answer: A — 35
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Q. In how many ways can 3 men and 2 women be arranged in a line if the men must be together?
Solution
Treat the 3 men as one unit. So, we have 3 units (MMM, W, W). The arrangements = 3! * 3! = 36.
Correct Answer: B — 120
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Q. In how many ways can 3 red balls and 2 blue balls be arranged in a row?
Solution
The total arrangements = 5! / (3! * 2!) = 10.
Correct Answer: A — 10
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Q. In how many ways can 3 red, 2 blue, and 1 green balls be arranged in a line?
Solution
The total arrangements = 6! / (3! * 2! * 1!) = 60.
Correct Answer: B — 120
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Q. In how many ways can 4 different books be chosen from a shelf of 10 books?
-
A.
210
-
B.
120
-
C.
240
-
D.
300
Solution
The number of ways to choose 4 books from 10 is C(10, 4) = 210.
Correct Answer: A — 210
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Q. In how many ways can 4 different books be selected from a shelf of 10 books?
-
A.
210
-
B.
120
-
C.
240
-
D.
300
Solution
The number of ways to choose 4 books from 10 is C(10,4) = 210.
Correct Answer: A — 210
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Q. In how many ways can 4 different colored balls be arranged in a line?
Solution
The number of arrangements is 4! = 24.
Correct Answer: B — 24
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Q. In how many ways can 4 different colored balls be placed in 3 different boxes?
Solution
Each ball can go into any of the 3 boxes, so the total ways = 3^4 = 81.
Correct Answer: A — 81
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Q. In how many ways can 4 different prizes be awarded to 3 students?
Solution
The number of ways is P(4, 3) = 4! / 1! = 24.
Correct Answer: C — 36
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Q. In how many ways can 4 different prizes be distributed among 3 students?
Solution
Each prize can go to any of the 3 students, so the total ways = 3^4 = 81.
Correct Answer: A — 81
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Q. In how many ways can 4 students be selected from a group of 10?
-
A.
210
-
B.
120
-
C.
240
-
D.
300
Solution
The number of ways to select 4 students from 10 is C(10,4) = 210.
Correct Answer: A — 210
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Q. In how many ways can 5 different colored balls be arranged in a box?
-
A.
60
-
B.
120
-
C.
100
-
D.
80
Solution
The number of arrangements is 5! = 120.
Correct Answer: B — 120
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Q. In how many ways can 5 different flags be arranged on a pole?
Solution
The number of arrangements of 5 different flags is 5! = 120.
Correct Answer: A — 120
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Q. In how many ways can 5 different items be selected from 10 items?
-
A.
252
-
B.
120
-
C.
200
-
D.
300
Solution
The number of ways to select 5 items from 10 is C(10, 5) = 252.
Correct Answer: A — 252
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Q. In how many ways can 5 different objects be selected from 10 objects?
-
A.
252
-
B.
120
-
C.
10
-
D.
100
Solution
The number of ways to select 5 objects from 10 is 10C5 = 10! / (5! * 5!) = 252.
Correct Answer: A — 252
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Q. In how many ways can 6 different objects be selected and arranged in a line?
-
A.
720
-
B.
600
-
C.
840
-
D.
960
Solution
The number of arrangements of 6 different objects is 6! = 720.
Correct Answer: A — 720
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Q. In how many ways can 6 people be divided into 2 groups of 3?
Solution
The number of ways to divide 6 people into 2 groups of 3 is 6! / (3! * 3! * 2!) = 20.
Correct Answer: A — 20
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Q. In how many ways can 7 different items be selected and arranged in a line?
-
A.
5040
-
B.
720
-
C.
40320
-
D.
10080
Solution
The number of arrangements of 7 different items is 7! = 5040.
Correct Answer: C — 40320
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Q. In how many ways can 7 different objects be arranged in a circle?
-
A.
720
-
B.
5040
-
C.
7200
-
D.
600
Solution
The number of arrangements in a circle is (n-1)! = 6! = 720.
Correct Answer: B — 5040
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Q. In how many ways can the letters of the word 'LEVEL' be arranged?
Solution
The word 'LEVEL' has 5 letters with 'L' and 'E' repeating. The arrangements = 5! / (2! * 2!) = 30.
Correct Answer: B — 30
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Q. In the expansion of (1 + x)^10, what is the coefficient of x^5?
-
A.
252
-
B.
210
-
C.
120
-
D.
300
Solution
The coefficient of x^5 is C(10,5) = 10! / (5!5!) = 252.
Correct Answer: A — 252
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Q. In the expansion of (2 + 3x)^4, what is the coefficient of x?
Solution
The coefficient of x is C(4,1) * 2^3 * 3 = 4 * 8 * 3 = 96.
Correct Answer: A — 12
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Q. In the expansion of (2 + 3x)^4, what is the coefficient of x^2?
Solution
The coefficient of x^2 is C(4,2) * (2)^2 * (3)^2 = 6 * 4 * 9 = 216.
Correct Answer: B — 54
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Q. In the expansion of (2x + 3)^4, what is the coefficient of x^1?
Solution
The coefficient of x^1 is C(4,1) * (2)^1 * (3)^3 = 4 * 2 * 27 = 216.
Correct Answer: B — 48
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