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 determine how many houses have only a TV and a washing machine but not an AC, we need to apply the principle of inclusion-exclusion (PIE) for three sets. Let:

• T = 862 (houses with a TV)
• A = 783 (houses with an AC)
• W = 736 (houses with a washing machine)
• x = 95 (houses with only a TV)
• y = 136 (houses with only an AC)
• z = 88 (houses with only a washing machine)
• t = 398 (houses with all three appliances)

We need to find the number of houses that have only a TV and washing machine but not AC. Let this be denoted by b_TW.

Applying the principle of inclusion-exclusion for three sets, we have:

T + A + W - (TW + AW + AT) + t = 1500.

We know:

• TW = Houses with both TV & Washing Machine (including those with all three)
• AW = Houses with both AC & Washing Machine (including those with all three)
• AT = Houses with both AC & TV (including those with all three)

From the given data:

TW = T + W - x - z - 398

The number above are inclusive of the TVs and Washing Machines that might be only owned together:

• TW = 862 + 736 - 95 - 88 - 398 = 1017

This count TW includes the 398 houses with all three items. Therefore, let TW' = TW - t (houses with only TV and Washing Machine).

Therefore:

b_TW = TW - t = 1017 - 398 = 213.

Hence, the number of houses that own only a TV and a washing machine but not an AC is 213.