Question

The 38 movies in the video store fall into the following three categories: 10 action, 20 drama, and 18 comedy. However, some movies are classified under more than one category: 5 are both action and drama, 3 are both action and comedy and 4 are both drama and comedy. How many action-drama-comedy movies are there?

Answer 1

Hint: Here in this question, we have to find how many movies which belong to all the categories like action-drama-comedy in the video store. To find this we can use the method of Venn diagram and using basic arithmetic operation addition on simplification we get the required solution.

Complete step by step solution:
Venn Diagram is a diagram representing mathematical or logical sets pictorially as circles or closed curves within an enclosing rectangle (the universal set), common elements of the sets being represented by intersections of the circles. To draw a Venn diagram, first, the universal set should be decided. Now, every set is the subset of the universal set (U). This means that every other set will be inside the rectangle which represents the universal set. So, any set A (shaded region) will be represented as follows: Where U is a universal set.
Consider the given question:
Let's consider the movies belonging to the category action-drama-comedy be \[x\] .
Both action and drama category is \[5 - x\]
Both action and comedy category is \[3 - x\]
Both drama and comedy category is \[4 - x\]
The only action category movie is \[10 - \left( {5 + 3 - x} \right)\] \[ \Rightarrow \,\,10 - \left( {8 - x} \right)\] \[ \Rightarrow \,\,10 - 8 + x\] \[ \Rightarrow \,\,2 + x\]
The only drama category movie is \[20 - \left( {5 + 4 - x} \right)\] \[ \Rightarrow \,\,20 - \left( {9 - x} \right)\] \[ \Rightarrow \,\,20 - 9 + x\] \[ \Rightarrow \,\,11 + x\]
The only comedy category movie is \[18 - \left( {4 + 3 - x} \right)\] \[ \Rightarrow \,\,18 - \left( {7 - x} \right)\] \[ \Rightarrow \,\,18 - 7 + x\] \[ \Rightarrow \,\,11 + x\]
The Venn diagram for these data are shown below:

The total movies in video store is 38 means the sum of all category movies should be equal to 38, then
 \[ \Rightarrow \,\,x + \left( {5 - x} \right) + \left( {3 - x} \right) + \left( {4 - x} \right) + \left( {2 + x} \right) + \left( {11 + x} \right) + \left( {11 + x} \right) = 38\]
On simplification, we get
 \[ \Rightarrow \,\,x + \left( {5 - x} \right) + \left( {3 - x} \right) + \left( {4 - x} \right) + \left( {2 + x} \right) + \left( {11 + x} \right) + \left( {11 + x} \right) = 38\]
 \[ \Rightarrow \,\,36 + x = 38\]
Subtract both side by 36, the
 \[ \Rightarrow \,\,36 + x - 36 = 38 - 36\]
 \[ \Rightarrow \,\,x = 2\]
Hence, the action-drama-comedy category movies is 2.
So, the correct answer is “x = 2”.

Note: The question is belonging to the concept of set. Writing the data in the form of a Venn diagram will make it easy to analyse the given information and we get a clear picture of how many movies. While simplifying we use the simple arithmetic operations and hence we obtain the result.