Question:
In a beauty contest, half the number of experts voted for Miss. A and two thirds voted for Miss B, 10 voted for both and '6' did not vote for either. Then how many experts were there in all is
In a beauty contest, half the number of experts voted for Miss. A and two thirds voted for Miss B, 10 voted for both and '6' did not vote for either. Then how many experts were there in all is
Updated On: Jul 6, 2022
- 20
- 24
- 22
- 26
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The Correct Option is B
Solution and Explanation
Let the number of experts = x
$n\left(A\right) = \frac{x}{2}.n\left(B\right)= \frac{2x}{3} $
$ n\left(A\cap B\right) = 10, n\left(A^{C}\cap B^{C}\right) = 6 $
$n\left(A\cup B\right) = \frac{x}{2} + \frac{2x}{3} - 10 $
$n\left(A^{C}\cap B^{C}\right) = 6 \Rightarrow n\left(\mu\right) -n\left(A\cup B\right) = 6$
$ \Rightarrow x - \frac{x}{2} - \frac{2x}{3} + 10 = 6$
$ \Rightarrow 6x -3x -4x + 60 = 36$
$ \Rightarrow x = 24 $
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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 "|".