To find: Number of student taking neither tea nor coffee i.e., we have to find n(T’ ∩ C’). n(T’∩C’) = n(T∪C)’
n(U) – n(T∪C)
n(U) – [n(T) + n(C) – n(T∩C)]
600 – [150 + 225 – 100]
600 – 275
Hence, 325 students were taking neither tea nor coffee.
In a group of students 100 students know Hindi, 50 know English and 25 know both. Each of the students knows either Hindi or English. How many students are there in the group?
Let U be the set of all students in the group.
Let E be the set of all students who know English. Let H be the set of all students who know Hindi.
∴ H ∪ E = U
Accordingly, n(H) = 100 and n(E) = 50
n(U) = n(H) + – n(H ∩ E)
100 + 50 – 25
Hence, there are 125 students in the group.
In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I,11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
the number of people who read at least one of the newspapers.
the number of people who read exactly one newspaper. Answer
Let A be the set of people who read newspaper H. Let B be the set of people who read newspaper T. Let C be the set of people who read newspaper I. Accordingly, n(A) = 25, n(B) = 26, and n(C) = 26
Hence, 52 people read at least one of the newspapers.
(ii) Let a be the number of people who read newspapers H and T only.
Let b denote the number of people who read newspapers I and H only. Let c denote the number of people who read newspapers T and I only. Let d denote the number of people who read all three newspapers.
Accordingly, d = n(A ∩ B ∩ C) = 3 Now, n(A ∩ B) = a + d
n(B∩C) = c + d n(C∩A) = b + d
∴ a + d + c + d + b + d = 11 + 8 + 9 = 28
⇒ a + b + c + d = 28 – 2d = 28 – 6 = 22
Hence, (52 – 22) = 30 people read exactly one newspaper.
In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked product C only.
Let A, B, and C be the set of people who like product A, product B, and product C respectively.