Question:
In a class of $140$ students numbered $1$ to $140$, all even numbered students opted mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is :
In a class of $140$ students numbered $1$ to $140$, all even numbered students opted mathematics course, those whose number is divisible by $3$ opted Physics course and those whose number is divisible by $5$ opted Chemistry course. Then the number of students who did not opt for any of the three courses is :
Updated On: Apr 2, 2026
- 102
- 42
- 1
- 38
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The Correct Option is D
Solution and Explanation
Let $n(A)$ = number of students opted
Mathematics = $70$,
$n(B)$ = number of students opted Physics = $46$,
$n(C)$ = number of students opted Chemistry
= 28,
$n(A \cap B) = 23,$
$ n(B \cap C) = 9,$
$n(A \cap C) = 14,$
$n(A \cap B \cap C) = 4,$
Now $n(A \cup B \cup C)$
$= n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C)$
$- n(A \cap C) + n(A \cap B \cap C)$
$= 70 + 46 + 28 - 23 - 9 - 14 + 4 = 102 $
So number of students not opted for any course
= Total - $n(A \cup B \cup C)$
$= 140 - 102 = 38$
Mathematics = $70$,
$n(B)$ = number of students opted Physics = $46$,
$n(C)$ = number of students opted Chemistry
= 28,
$n(A \cap B) = 23,$
$ n(B \cap C) = 9,$
$n(A \cap C) = 14,$
$n(A \cap B \cap C) = 4,$
Now $n(A \cup B \cup C)$
$= n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C)$
$- n(A \cap C) + n(A \cap B \cap C)$
$= 70 + 46 + 28 - 23 - 9 - 14 + 4 = 102 $
So number of students not opted for any course
= Total - $n(A \cup B \cup C)$
$= 140 - 102 = 38$
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".