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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Q. In the expansion of (2x + 3)^4, what is the coefficient of x^2?
-
A.
108
-
B.
72
-
C.
36
-
D.
144
Solution
The coefficient of x^2 is C(4,2) * (2)^2 * (3)^2 = 6 * 4 * 9 = 216.
Correct Answer: B — 72
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Q. In the expansion of (2x + 3)^6, what is the coefficient of x^4?
-
A.
540
-
B.
720
-
C.
810
-
D.
900
Solution
The coefficient of x^4 is C(6,4) * (2)^4 * (3)^2 = 15 * 16 * 9 = 2160.
Correct Answer: B — 720
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Q. In the expansion of (2x + 3y)^4, what is the coefficient of x^2y^2?
-
A.
108
-
B.
72
-
C.
36
-
D.
144
Solution
The coefficient is C(4,2) * (2)^2 * (3)^2 = 6 * 4 * 9 = 216.
Correct Answer: A — 108
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Q. In the expansion of (2x + 5)^4, what is the coefficient of x^2?
-
A.
300
-
B.
600
-
C.
450
-
D.
500
Solution
The coefficient of x^2 is C(4,2) * (2)^2 * (5)^2 = 6 * 4 * 25 = 600.
Correct Answer: A — 300
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Q. In the expansion of (2x - 3)^3, what is the coefficient of x?
-
A.
-9
-
B.
-18
-
C.
-27
-
D.
-6
Solution
The coefficient of x is C(3,1) * (2)^1 * (-3)^2 = 3 * 2 * 9 = -54.
Correct Answer: B — -18
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Q. In the expansion of (2x - 3)^5, what is the coefficient of x^1?
-
A.
-15
-
B.
-30
-
C.
30
-
D.
15
Solution
The coefficient of x^1 is C(5,1) * (2)^1 * (-3)^4 = 5 * 2 * 81 = 810.
Correct Answer: B — -30
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Q. In the expansion of (2x - 3)^5, what is the coefficient of x^2?
-
A.
-90
-
B.
-120
-
C.
-150
-
D.
-180
Solution
The coefficient of x^2 is C(5,2) * (2)^2 * (-3)^3 = 10 * 4 * (-27) = -1080.
Correct Answer: B — -120
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Q. In the expansion of (2x - 3)^5, what is the coefficient of x^3?
-
A.
-540
-
B.
-720
-
C.
-960
-
D.
-1080
Solution
The coefficient of x^3 is C(5,3) * (2)^3 * (-3)^2 = 10 * 8 * 9 = -720.
Correct Answer: C — -960
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Q. In the expansion of (2x - 3)^5, what is the coefficient of x^4?
-
A.
-90
-
B.
-120
-
C.
-150
-
D.
-180
Solution
The coefficient of x^4 is C(5,4) * (2)^4 * (-3)^1 = 5 * 16 * (-3) = -240.
Correct Answer: B — -120
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Q. In the expansion of (x + 1/x)^6, what is the coefficient of x^0?
Solution
The coefficient of x^0 in (x + 1/x)^6 is given by 6C3 = 20.
Correct Answer: C — 20
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Q. In the expansion of (x + 1/x)^8, what is the coefficient of x^4?
Solution
The coefficient of x^4 is C(8,4) = 70.
Correct Answer: B — 56
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Q. In the expansion of (x + 2)^5, what is the coefficient of x^3?
Solution
The coefficient of x^3 is C(5,3) * (2)^2 = 10 * 4 = 40.
Correct Answer: B — 60
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Q. In the expansion of (x + 2)^6, what is the term containing x^4?
-
A.
240
-
B.
360
-
C.
720
-
D.
480
Solution
The term containing x^4 is C(6,4) * (2)^2 * (x)^4 = 15 * 4 * x^4 = 60x^4.
Correct Answer: C — 720
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Q. In the expansion of (x + 2)^7, what is the term containing x^5?
-
A.
672
-
B.
6720
-
C.
67200
-
D.
672000
Solution
The term containing x^5 is given by 7C5 * (2)^2 * (x)^5 = 21 * 4 * x^5 = 84x^5. Thus, the term is 84x^5.
Correct Answer: B — 6720
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Q. In the expansion of (x + 5)^6, what is the coefficient of x^3?
-
A.
150
-
B.
200
-
C.
250
-
D.
300
Solution
The coefficient of x^3 is C(6, 3) * 5^3 = 20 * 125 = 250.
Correct Answer: B — 200
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Q. In the expansion of (x - 2)^6, what is the coefficient of x^4?
Solution
The coefficient of x^4 is given by C(6,4) * (-2)^2 = 15 * 4 = 60.
Correct Answer: C — 90
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Q. In the expansion of (x - 2)^8, what is the coefficient of x^5?
-
A.
-2240
-
B.
-1280
-
C.
-1120
-
D.
-960
Solution
The coefficient of x^5 is C(8,5) * (-2)^3 = 56 * (-8) = -448.
Correct Answer: A — -2240
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Q. In triangle ABC, if a = 7, b = 24, and c = 25, what is the area of the triangle?
-
A.
84
-
B.
96
-
C.
120
-
D.
144
Solution
Using Heron's formula, s = (7 + 24 + 25)/2 = 28. Area = √[s(s-a)(s-b)(s-c)] = √[28(28-7)(28-24)(28-25)] = √[28*21*4*3] = 84.
Correct Answer: A — 84
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