Mathematics for Competitive Exam Success

Mathematics

3D Geometry Applications of Derivatives Binomial Theorem Circles Complex Numbers Conic Sections Coordinate Geometry Differential Equations Differentiation Integration Limits & Continuity Matrices & Determinants Permutations & Combinations Probability Quadratic Equations Sequences & Series Sets & Relations Straight Lines Trigonometry Vector Algebra

Q. How many ways can 4 different letters be chosen from the word 'COMPUTE'? (2022)

A. 35
B. 70
C. 56
D. 84

Solution

The number of ways is 8C4 = 70.

Correct Answer: C — 56

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Q. How many ways can 4 letters be chosen from the word 'COMBINATION'?

A. 210
B. 120
C. 150
D. 180

Solution

The number of ways to choose 4 letters from 11 is 11C4 = 330.

Correct Answer: A — 210

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Q. How many ways can 4 people be seated in a row of 6 chairs? (2021)

A. 360
B. 720
C. 240
D. 480

Solution

The number of ways is 6P4 = 6! / (6-4)! = 360.

Correct Answer: A — 360

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Q. How many ways can 4 prizes be distributed among 10 students if each student can receive at most one prize? (2019)

A. 5040
B. 720
C. 1000
D. 1200

Solution

The number of ways to distribute 4 prizes among 10 students is 10P4 = 5040.

Correct Answer: A — 5040

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Q. How many ways can 4 students be selected from a group of 12? (2014)

A. 495
B. 144
C. 120
D. 220

Solution

The number of ways to select 4 students from 12 is given by 12C4 = 495.

Correct Answer: A — 495

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Q. How many ways can 4 students be selected from a group of 15?

A. 1365
B. 455
C. 105
D. 210

Solution

The number of ways to select 4 students from 15 is given by 15C4 = 1365.

Correct Answer: A — 1365

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Q. How many ways can 5 different books be selected from a shelf of 10?

A. 252
B. 120
C. 300
D. 200

Solution

The number of ways to choose 5 from 10 is C(10,5) = 252.

Correct Answer: A — 252

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Q. How many ways can 5 different cards be chosen from a deck of 52 cards? (2019)

A. 2598960
B. 100
C. 1000
D. 500

Solution

The number of ways to choose 5 cards from 52 is given by 52C5 = 2598960.

Correct Answer: A — 2598960

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Q. How many ways can 5 different colored balls be arranged in a line? (2016)

A. 120
B. 60
C. 30
D. 24

Solution

The number of arrangements of 5 different colored balls is 5! = 120.

Correct Answer: A — 120

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Q. How many ways can 5 different letters be arranged if 2 letters are always together?

A. 48
B. 120
C. 240
D. 720

Solution

Treat the 2 letters as one unit. So, we have 4 units to arrange: 4! * 2! = 48.

Correct Answer: C — 240

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Q. How many ways can 5 different letters be arranged if 2 letters are identical? (2017)

A. 60
B. 120
C. 30
D. 20

Solution

The number of arrangements of 5 letters where 2 are identical is 5! / 2! = 60.

Correct Answer: B — 120

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Q. How many ways can 5 different letters be arranged if 2 letters must always be together?

A. 60
B. 120
C. 240
D. 720

Solution

Treat the 2 letters as one unit. So, we have 4 units to arrange: 4! * 2! = 48.

Correct Answer: C — 240

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Q. How many ways can 5 people be seated in a row of 5 chairs?

A. 120
B. 60
C. 30
D. 24

Solution

The number of arrangements of 5 people in 5 chairs is 5! = 120.

Correct Answer: A — 120

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Q. How many ways can 5 students be seated in a row of 8 chairs? (2016)

A. 6720
B. 120
C. 240
D. 360

Solution

The number of ways to seat 5 students in 8 chairs is 8P5 = 6720.

Correct Answer: A — 6720

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Q. How many ways can 6 different books be arranged on a shelf? (2017)

A. 720
B. 600
C. 840
D. 960

Solution

The number of arrangements of 6 different books is 6! = 720.

Correct Answer: A — 720

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Q. How many ways can 6 different flags be arranged on a pole? (2023)

A. 720
B. 600
C. 480
D. 360

Solution

The number of arrangements of 6 different flags is 6! = 720.

Correct Answer: A — 720

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Q. How many ways can 6 different letters be arranged if 2 letters are always together?

A. 120
B. 240
C. 720
D. 1440

Solution

Treat the 2 letters as one unit. So, we have 5 units to arrange: 5! = 120. The 2 letters can be arranged in 2! = 2 ways. Total = 120 * 2 = 240.

Correct Answer: D — 1440

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Q. How many ways can 6 people be seated in a row? (2017)

A. 720
B. 600
C. 480
D. 360

Solution

The number of arrangements of 6 people in a row is 6! = 720.

Correct Answer: A — 720

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Q. How many ways can 6 people be selected from a group of 10? (2020)

A. 210
B. 120
C. 300
D. 150

Solution

The number of ways to select 6 from 10 is C(10, 6) = 210.

Correct Answer: A — 210

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Q. How many ways can 7 different books be arranged on a shelf if 3 specific books must be together?

A. 720
B. 120
C. 5040
D. 840

Solution

Treat the 3 specific books as one unit. So, we have 5 units to arrange: 5! * 3! = 360.

Correct Answer: C — 5040

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Q. How many ways can 8 different books be arranged on a shelf if 3 specific books must be together?

A. 720
B. 5040
C. 40320
D. 2880

Solution

Treat the 3 specific books as one unit. So, we have 6 units to arrange: 6! = 720. The 3 books can be arranged in 3! = 6 ways. Total = 720 * 6 = 4320.

Correct Answer: B — 5040

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Q. How many ways can 8 different colored balls be arranged in a line?

A. 40320
B. 720
C. 1000
D. 100

Solution

The number of arrangements is 8! = 40320.

Correct Answer: A — 40320

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Q. How many ways can a committee of 3 be formed from 7 people? (2021)

A. 21
B. 35
C. 15
D. 28

Solution

The number of ways to form a committee of 3 from 7 is given by 7C3 = 35.

Correct Answer: B — 35

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Q. How many ways can a committee of 4 be formed from 10 people? (2019)

A. 210
B. 120
C. 100
D. 30

Solution

The number of ways to form a committee of 4 from 10 is given by 10C4 = 210.

Correct Answer: A — 210

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Q. How many ways can a committee of 4 be formed from 8 people? (2019)

A. 70
B. 80
C. 90
D. 100

Solution

The number of ways to form a committee of 4 from 8 is 8C4 = 70.

Correct Answer: A — 70

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Q. If 2x = 10, what is the value of x? (2015)

A. 5
B. 10
C. 2
D. 20

Solution

Dividing both sides by 2 gives x = 10/2 = 5.

Correct Answer: A — 5

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Q. If 3x = 12, what is the value of x? (2019)

A. 3
B. 4
C. 5
D. 2

Solution

x = 12 / 3 = 4.

Correct Answer: B — 4

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Q. If a = 2 and b = 3, what is the value of a^3 + b^3? (2021)

A. 35
B. 27
C. 10
D. 8

Solution

a^3 = 2^3 = 8 and b^3 = 3^3 = 27, so 8 + 27 = 35.

Correct Answer: A — 35

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Q. If A = 2i + 3j and B = 3i + 4j, what is the angle between A and B? (2021)

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 180 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.7071, θ = 45 degrees.

Correct Answer: B — 45 degrees

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Q. If A = 3i + 4j and B = 2i - j, what is A + B? (2021)

A. 5i + 3j
B. 5i + 5j
C. 1i + 5j
D. 1i + 3j

Solution

A + B = (3i + 4j) + (2i - j) = (3 + 2)i + (4 - 1)j = 5i + 3j.

Correct Answer: A — 5i + 3j

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Mathematics

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