Question

An electronic company conducts a survey of 1500 houses for their products. The survey suggested that 862 houses own TV, 783 houses has AC and 736 houses has washing machine. There were 95 houses having only TV, 136 houses having only AC and 88 houses having only washing machine. There were 398 houses having all the three equipments. How many houses have only TV and washing machine but not AC?

• A. 65
• B. 119
• C. 184
• D. 185
• E. 213

Answer 1

Answer: (E)

To solve this problem, we can use the principle of inclusion-exclusion in set theory. Let's denote:

• \(T\): the set of houses having TV. 
• \(A\): the set of houses having AC.
• \(W\): the set of houses having a washing machine.

From the information given:

• \(|T| = 862\)
• \(|A| = 783\)
• \(|W| = 736\)
• \(|T \cap A \cap W| = 398\)
• Houses with only TV = \(95\)
• Houses with only AC = \(136\)
• Houses with only washing machine = \(88\)

We want to find the number of houses that have only TV and washing machine, but not AC. This corresponds to the set \(|T \cap W \cap \overline{A}|\).

We can determine the union of sets using the formula:

• \(|T \cup A \cup W| = |T| + |A| + |W| - |T \cap A| - |A \cap W| - |T \cap W| + |T \cap A \cap W|\)

First, let's find \(|T \cap A|\), \(|A \cap W|\), \(|T \cap W|\) as follows:

• \(|T \cap A| = |T| - (only \, TV) - |T \cap W| - |T \cap A \cap W|\)
• \(|A \cap W| = |A| - (only \, AC) - |T \cap A \cap W|\)
• \(|T \cap W| = |W| - (only \, washing \, machine) - |T \cap A \cap W|\)

Next, use known values to find:

• \(|T \cap A| = 862 - 95 - x - 398\)
• \(|A \cap W| = 783 - 136 - 398\) = 249
• \(|T \cap W| = 736 - 88 - 398\) = 250

Now, use the principle of inclusion-exclusion to find the unknown value \(x\):

• Calculate the total number of houses having only TV and washing machine, but not AC: \(|T \cap W \cap \overline{A}| = |T \cap W| - |T \cap A \cap W| = 250 - 37\) = 213.

Thus, the number of houses having only TV and washing machine, but not AC is 213.