IPMPrepA set has 6 elements. How many of its subsets have at least 2 elements?
Solution
Total subsets = 26 = 64.
Remove the empty set and the 6 single-element subsets: 64 − 1 − 6 = 57.
Answer: 57
Quantitative Ability · Algebra
Sets and Venn Diagrams: IPMAT practice questions
24 questions, 15 of them short answer (type the number, as in IPMAT Indore). Each is marked with the exam level it suits. Answer, then open the step-by-step solution.
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In a survey of 101 people, 48 read paper A, 32 read B and 32 read C; 15 read A and B, 12 read B and C, 16 read C and A, and 8 read all three. How many read exactly one paper?
Solution
n(A ∪ B ∪ C) = 48 + 32 + 32 − 15 − 12 − 16 + 8 = 77.
Exactly one = ΣA − 2Σ(pairs) + 3(all three) = 112 − 86 + 24 = 50.
Answer: (C) 50
In a survey of 74 people, 31 read paper A, 23 read B and 36 read C; 11 read A and B, 9 read B and C, 11 read C and A, and 4 read all three. How many read none of the three papers?
Solution
n(A ∪ B ∪ C) = 31 + 23 + 36 − 11 − 9 − 11 + 4 = 63.
None = 74 − 63 = 11.
Answer: 11
How many integers from 1 to 500 are divisible by at least one of 3, 5 and 7?
Solution
Inclusion–exclusion: 166 + 100 + 71 − 33 − 14 − 23 + 4 = 271 are divisible by at least one.
Answer: 271
How many integers from 1 to 500 are divisible by none of 2, 5 and 7?
Solution
Inclusion–exclusion: 250 + 100 + 71 − 50 − 14 − 35 + 7 = 329 are divisible by at least one.
Divisible by none: 500 − 329 = 171.
Answer: (C) 171
Of 120 students, 111 like Maths and 84 like English. What is the greatest possible number who like both?
Solution
Both can be at most the smaller group, 84, when every English fan likes the other too.
So the answer is 84.
Answer: (A) 84
Of 120 students, 95 like Maths and 78 like English. What is the least possible number who like both?
Solution
Both ≥ 95 + 78 − 120 = 53 (at most 120 students in all).
So the answer is 53.
Answer: 53
Of 200 students, 190 like Maths and 166 like English. What is the greatest possible number who like both?
Solution
Both can be at most the smaller group, 166, when every English fan likes the other too.
So the answer is 166.
Answer: (A) 166
In a survey of 97 people, 54 read paper A, 41 read B and 23 read C; 14 read A and B, 11 read B and C, 10 read C and A, and 4 read all three. How many read none of the three papers?
Solution
n(A ∪ B ∪ C) = 54 + 41 + 23 − 14 − 11 − 10 + 4 = 87.
None = 97 − 87 = 10.
Answer: 10
In a survey of 76 people, 33 read paper A, 26 read B and 38 read C; 6 read A and B, 13 read B and C, 6 read C and A, and 4 read all three. How many read none of the three papers?
Solution
n(A ∪ B ∪ C) = 33 + 26 + 38 − 6 − 13 − 6 + 4 = 76.
None = 76 − 76 = 0.
Answer: 0
A set has 9 elements. How many of its subsets have at least 2 elements?
Solution
Total subsets = 29 = 512.
Remove the empty set and the 9 single-element subsets: 512 − 1 − 9 = 502.
Answer: 502
n(A) = 4 and n(B) = 3. How many relations are there from A to B?
Solution
A relation is any subset of A × B, which has 4 × 3 = 12 elements.
Number of subsets = 212 = 4096.
Answer: 4096
How many integers from 1 to 100 are divisible by none of 2, 5 and 7?
Solution
Inclusion–exclusion: 50 + 20 + 14 − 10 − 2 − 7 + 1 = 66 are divisible by at least one.
Divisible by none: 100 − 66 = 34.
Answer: 34
A set has 8 elements. How many of its subsets have at least 2 elements?
Solution
Total subsets = 28 = 256.
Remove the empty set and the 8 single-element subsets: 256 − 1 − 8 = 247.
Answer: (D) 247
In a survey of 122 people, 42 read paper A, 48 read B and 57 read C; 9 read A and B, 14 read B and C, 13 read C and A, and 2 read all three. How many read exactly one paper?
Solution
n(A ∪ B ∪ C) = 42 + 48 + 57 − 9 − 14 − 13 + 2 = 113.
Exactly one = ΣA − 2Σ(pairs) + 3(all three) = 147 − 72 + 6 = 81.
Answer: (A) 81
How many integers from 1 to 1000 are divisible by at least one of 3, 5 and 7?
Solution
Inclusion–exclusion: 333 + 200 + 142 − 66 − 28 − 47 + 9 = 543 are divisible by at least one.
Answer: 543
How many integers from 1 to 500 are divisible by none of 4, 6 and 9?
Solution
Inclusion–exclusion: 125 + 83 + 55 − 41 − 27 − 13 + 13 = 195 are divisible by at least one.
Divisible by none: 500 − 195 = 305.
Answer: (A) 305
How many integers from 1 to 200 are divisible by none of 2, 3 and 7?
Solution
Inclusion–exclusion: 100 + 66 + 28 − 33 − 9 − 14 + 4 = 142 are divisible by at least one.
Divisible by none: 200 − 142 = 58.
Answer: (D) 58
In a survey of 108 people, 44 read paper A, 46 read B and 54 read C; 16 read A and B, 17 read B and C, 11 read C and A, and 7 read all three. How many read exactly two papers?
Solution
n(A ∪ B ∪ C) = 44 + 46 + 54 − 16 − 17 − 11 + 7 = 107.
Exactly two = Σ(pairs) − 3(all three) = 44 − 21 = 23.
Answer: 23
Of 200 students, 121 like Hindi films and 141 like English films. What is the least possible number who like both?
Solution
Both ≥ 121 + 141 − 200 = 62 (at most 200 students in all).
So the answer is 62.
Answer: 62
A = {1, 2, …, 9}. How many subsets of A contain all of 1, 2, …, 7?
Solution
The 7 required elements are fixed; each of the other 2 can be in or out.
22 = 4.
Answer: 4
A = {1, 2, …, 4}. How many subsets of A contain the element 1?
Solution
The 1 required element is fixed; each of the other 3 can be in or out.
23 = 8.
Answer: 8
Of 80 students, 46 like cricket and 62 like football. What is the least possible number who like both?
Solution
Both ≥ 46 + 62 − 80 = 28 (at most 80 students in all).
So the answer is 28.
Answer: 28
Of 120 students, 76 like tea and 107 like coffee. What is the greatest possible number who like both?
Solution
Both can be at most the smaller group, 76, when every tea fan likes the other too.
So the answer is 76.
Answer: (A) 76
These are original practice questions written by Target Campus in the pattern, syllabus and difficulty of IPMAT Indore, IPMAT Rohtak, JIPMAT, NPAT and SET. They are not previous-year papers.