Questions & explanations
1. In a survey of 200 people, 80 read magazine A, 90 read magazine B, 70 read magazine C, 30 read both A and B, 25 read both A and C, 20 read both B and C, and 10 read all three. How many read exactly one magazine?
- 100
- 110
- 120
- 130
- 140
Answer: 120
Use inclusion-exclusion: |A ∪ B ∪ C| = 80+90+70 - (30+25+20) + 10 = 240 - 75 + 10 = 175. Number reading exactly one = |A|+|B|+|C| - 2*(sum of pairwise intersections) + 3*|triple| = 240 - 2*75 + 3*10 = 240 - 150 + 30 = 120. Alternatively, compute each: A only = 80 - (30+25) + 10 = 35; B only = 90 - (30+20) + 10 = 50; C only = 70 - (25+20) + 10 = 35; sum = 120.
2. A box contains 4 red, 3 blue, and 2 green marbles. If 4 marbles are drawn at random without replacement, what is the probability that there is at least one marble of each color?
- 4/7
- 12/21
- 2/3
- 5/9
- 8/21
Answer: 4/7
Total ways: C(9,4)=126. Favorable: must have at least one of each color. Cases: (2,1,1) distribution. Number of ways: choose which color has 2: 3 choices. For red2: C(4,2)*C(3,1)*C(2,1)=6*3*2=36; blue2: C(4,1)*C(3,2)*C(2,1)=4*3*2=24; green2: C(4,1)*C(3,1)*C(2,2)=4*3*1=12. Total favorable = 36+24+12=72. Probability = 72/126 = 4/7.
3. A committee of 4 people is to be chosen from a group of 6 men and 4 women. How many committees have at least 2 women?
- 115
- 185
- 195
- 210
- 230
Answer: 185
Total committees: C(10,4)=210. Committees with 0 women: C(6,4)=15. With 1 woman: C(4,1)*C(6,3)=4*20=80. At least 2 women = total - (0+1) = 210 - 95 = 115. Alternatively, sum: 2 women: C(4,2)*C(6,2)=6*15=90; 3 women: C(4,3)*C(6,1)=4*6=24; 4 women: C(4,4)*C(6,0)=1*1=1; total=115.
4. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set. Which of the following statements is true?
- The set {1, 2, 3} is a subset of U.
- The set {1, 2, 3} is an element of U.
- The set {1, 2, 3} is the universal set.
- The set {1, 2, 3} is equal to U.
- The set {1, 2, 3} is not a subset of U.
Answer: The set {1, 2, 3} is a subset of U.
A set A is a subset of set B if every element of A is also an element of B. Here, every element of {1,2,3} is in U, so it is a subset. The other options are false because {1,2,3} is not an element (elements are numbers), not equal to U, and not the universal set.
5. In how many ways can 6 people be seated around a circular table if two particular people must sit next to each other?
- 48
- 120
- 240
- 720
- 144
Answer: 48
Treat the two people as a block. For circular arrangements, number of ways to arrange n distinct objects is (n-1)!. Here we have 5 objects (block + 4 individuals): (5-1)! = 4! = 24. Within the block, the two can be arranged in 2! = 2 ways. Total = 24 * 2 = 48.
6. How many ways are there to distribute 10 identical candies to 4 distinct children, if each child must receive at least one candy?
- 84
- 120
- 126
- 210
- 220
Answer: 84
This is a stars and bars problem with the constraint that each child gets at least one. Give each child 1 candy first, leaving 6 candies to distribute among 4 children without restriction. Number of ways = C(6+4-1,4-1) = C(9,3) = 84.
7. In how many ways can 10 identical candies be distributed among 4 distinct children, if each child must receive at least one candy?
- 84
- 120
- 126
- 210
- 220
Answer: 84
This is combinations with repetition (stars and bars) with lower bound. Give each child 1 candy first, leaving 6 candies. Number of ways to distribute 6 identical candies among 4 children = C(6+4-1,4-1) = C(9,3) = 84.
8. Let the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}. What is the complement of A, denoted A'?
- {1, 3, 5, 7, 9}
- {2, 4, 6, 8, 10}
- {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- {}
- {1, 2, 3, 4, 5}
Answer: {1, 3, 5, 7, 9}
The complement of a set A relative to the universal set U is the set of all elements in U that are not in A. Since A contains all even numbers from 1 to 10, its complement is the odd numbers: {1,3,5,7,9}.
9. A committee of 4 people is to be selected from a group of 5 men and 4 women. How many committees can be formed if the committee must include at least one man and at least one woman?
- 120
- 125
- 130
- 135
- 140
Answer: 120
Total committees without restriction: C(9,4)=126. Committees with no men (all women): C(4,4)=1. Committees with no women (all men): C(5,4)=5. So committees with at least one of each = 126 - 1 - 5 = 120.
10. A committee of 4 members is to be formed from a group of 10 people, and then the committee must select a chairperson, a vice-chairperson, a secretary, and a treasurer. How many different ways can this be done?
- 5040
- 151200
- 210
- 50400
- 30240
Answer: 5040
First, choose 4 people from 10: C(10,4) = 210. Then assign 4 distinct roles to these 4 people: 4! = 24. Total = 210 × 24 = 5040. Alternatively, directly permute 4 from 10: P(10,4) = 10×9×8×7 = 5040.
11. In a survey of 100 students, 60 like math, 50 like science, and 30 like both. How many students like neither math nor science?
- 10
- 20
- 30
- 40
- 50
Answer: 20
Let M = math, S = science. |M| = 60, |S| = 50, |M ∩ S| = 30. Number liking at least one = |M ∪ S| = |M| + |S| - |M ∩ S| = 60 + 50 - 30 = 80. Total students = 100, so neither = 100 - 80 = 20.
12. A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If two marbles are drawn without replacement, what is the probability that at least one is red or at least one is blue?
- 1/45
- 44/45
- 7/9
- 8/9
- 1
Answer: 44/45
Total marbles = 10. P(at least one red or at least one blue) = 1 - P(neither red nor blue) = 1 - P(both green). P(both green) = (2/10)*(1/9) = 2/90 = 1/45. So probability = 1 - 1/45 = 44/45.