Question:
In a class of 80 students numbered a to 80, all odd numbered students opt of Cricket, students whose numbers are divisible by 5 opt for Football and those whose numbers are divisible by 7 opt for Hockey. The number of students who do not opt any of the three games, is
In a class of 80 students numbered a to 80, all odd numbered students opt of Cricket, students whose numbers are divisible by 5 opt for Football and those whose numbers are divisible by 7 opt for Hockey. The number of students who do not opt any of the three games, is
Updated On: Jul 5, 2022
- 13
- 24
- 28
- 52
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The Correct Option is C
Solution and Explanation
Numbers which are divisible by 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80 they are 16 in numbers. Now, Numbers which are divisible by 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77 they are 11 in numbers.
Also, total odd numbers = 40
Let C represents the students who opt. for cricket, F
for football and H for hockey.
$\therefore$ we have $n\left(C\right) = 40, n\left(F\right) = 16, n\left(H\right) = 11$
Now, $C \cap F =$ Odd numbers which are divisible by 5.
$C\cap H =$ Odd numbers which are divisible by 7.
$F \cap H =$ Numbers which are divisible by both 5 and 7.
$n\left(C \cap F\right), 8, n\left(C \cap H\right) = 6$,
$n\left(F\cap H\right) = 2, n \left(C\cap F \cap H\right) = 1$
We Know
$n\left(C\cup F\cup H\right) = n\left(C\right) + n\left(F\right) + n\left(H\right) - n\left(C \cap F\right) - n\left(C \cap H\right)$
$- n\left(F \cap H\right) + n\left(C \cap H \cap F\right)$
$n\left(C\cup F\cup H\right) = 67 - 16 + 1 = 52$
$\therefore n\left(C' \cap F' \cap H'\right)$
= Total students $- n\left(C \cup F \cup H\right)$
$n\left(C' \cap F'\cap H'\right)= 80 - 52 = 28$
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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 "|".