Question 1
The letters of the word VIOLENT are arranged so that the vowels occupy even place only. The number of permutations is:
A
144✓✗
B
120✓✗
C
24✓✗
D
72✓✗
Explanation
VIOLENT has 7 letters: 3 vowels (I, O, E) and 4 consonants (V, L, N, T).
Of the 7 positions, exactly 3 are even (2nd, 4th, 6th) -- matching the 3 vowels exactly.
Vowels arranged in the 3 even positions: 3! = 6 ways
Consonants arranged in the remaining 4 (odd) positions: 4! = 24 ways
Total = 3! × 4! = 6 × 24 = 144
Question 2
A garden has 6 tall trees in a row. In how many ways can 5 children stand, one in a gap between the trees, in order to pose for a photograph?
A
24✓✗
B
120✓✗
C
720✓✗
D
30✓✗
Explanation
6 trees in a row create 5 gaps between them.
5 children are to be arranged in these 5 gaps: 5! = 120
Question 3
Find the number of arrangements in which the letters of the word MONDAY be arranged so that the words thus formed begin with M and do not end with N.
A
720✓✗
B
120✓✗
C
96✓✗
D
None✓✗
Explanation
MONDAY has 6 distinct letters.
Arrangements beginning with M: fix M first, arrange remaining 5 letters (O,N,D,A,Y) → 5! = 120
Of these, arrangements that ALSO end with N: fix M first and N last, arrange remaining 4 letters (O,D,A,Y) in between → 4! = 24
Required arrangements = 120 - 24 = 96
Question 4
Five bulbs, of which three are defective, are to be tried in two light-points in a dark room. In how many trials shall the room necessarily be lightened?
A
10✓✗
B
7✓✗
C
3✓✗
D
None of these✓✗
Explanation
Total ways to pick 2 bulbs out of 5 = C(5,2) = 10
Ways where BOTH picked bulbs are defective (room stays dark) = C(3,2) = 3
Ways the room is lightened (at least one working bulb) = 10 - 3 = 7
Question 5
The number of ways of painting the faces of a cube by 6 different colours is:
A
30✓✗
B
36✓✗
C
24✓✗
D
1✓✗
Explanation
With all 6 faces getting distinct colours, the number of ways to assign 6 colours to 6 faces (ignoring orientation) is 6! = 720.
A cube has 24 rotational symmetries, so each distinct painting is counted 24 times.
Distinct paintings = 720 / 24 = 30
Question 6
If from a population with 25 members, a random sample without replacement of 2 members is taken, the number of all such samples is:
A
300✓✗
B
625✓✗
C
50✓✗
D
600✓✗
Explanation
Number of samples = C(25,2) = 25 × 242 = 300
Question 7
A room has 10 doors. In how many ways can a man enter the room by one door and come out by a different door?
A
90✓✗
B
100✓✗
C
50✓✗
D
None of these✓✗
Explanation
He can enter by any of 10 doors, then exit by any of the remaining 9 doors.
Total ways = 10 × 9 = 90
Question 8
There are 12 questions to be answered as Yes or No. In how many ways can this be answered?
A
1021✓✗
B
2048✓✗
C
4096✓✗
D
None of the above✓✗
Explanation
Each question has 2 possible answers, independently, over 12 questions:
212 = 4096
Question 9
In how many ways can 3 prizes out of 5 be distributed amongst 3 brothers equally?
A
10✓✗
B
45✓✗
C
60✓✗
D
120✓✗
Explanation
First choose which 3 of the 5 prizes are distributed: C(5,3) = 10
Then distribute those 3 prizes among 3 brothers, one each: 3! = 6
Total = 10 × 6 = 60
Question 10
A box contains 7 red, 6 white and 4 blue balls. How many selections of three balls can be made so that none is red?
A
90✓✗
B
120✓✗
C
48✓✗
D
None of these✓✗
Explanation
Non-red balls = 6 white + 4 blue = 10.
Ways to select 3 from these 10 = C(10,3) = 120
Question 11
A user wants to create a password using 4 lowercase letters (a-z) and 3 uppercase letters (A-Z). No letter can be repeated in any form. In how many ways can the password be created if the password must start with an uppercase letter?
A
26 × 25 × 24 × 23 × 22 × 5 × 21✓✗
B
26 × 25 × 24 × 23 × 22 × 2 × 21✓✗
C
26 × 5 × 25 × 24 × 23 × 2 × 22 × 21✓✗
D
6 × 26 × 25 × 24 × 23 × 22 × 21✓✗
Explanation
The first character must be an uppercase letter: 26 choices.
The remaining 6 positions are filled with the remaining letters (2 more uppercase from 25 remaining, and 4 lowercase from 26), with no repetition -- worked out position by position as a sequential count of remaining choices.
Question 12
A boy has 3 library tickets and 8 books of his interest in the library. Of these 8, he does not want to borrow Mathematics Part-II unless Mathematics Part-I is also borrowed. In how many ways can he choose the three books to be borrowed?
A
41✓✗
B
51✓✗
C
61✓✗
D
71✓✗
Explanation
Total ways to choose 3 books from 8 = C(8,3) = 56
Invalid cases (Part-II chosen but Part-I not): choose Part-II, then the remaining 2 books from the other 6 (excluding both Part-I and Part-II) = C(6,2) = 15
Valid ways = 56 - 15 = 41
Question 13
5 persons are sitting at a round table in such a way that the tallest person is always on the right side of the shortest person. The number of such arrangements is
A
6✓✗
B
8✓✗
C
24✓✗
D
none of these✓✗
Explanation
Treat the tallest and shortest persons as one unit (in a fixed relative order) along with the remaining 3 persons: circular arrangements of these 4 units = (4 - 1)! = 6
Question 14
An examination paper with 10 questions consists of 6 questions in Algebra and 4 questions in Geometry. At least one question from each section is to be attempted. In how many ways can this be done?
A
945✓✗
B
100✓✗
C
1000✓✗
D
none of these✓✗
Explanation
Ways to choose at least one question from Algebra = 26 - 1 = 63
Ways to choose at least one question from Geometry = 24 - 1 = 15
Total ways = 63 × 15 = 945
Question 15
If 12 school teams are participating in a quiz contest, then the number of ways the first, second and third positions may be won is
A
1,230✓✗
B
1,320✓✗
C
3,210✓✗
D
none of these✓✗
Explanation
Number of ways = 12 × 11 × 10 = 1,320
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