Set Theory Principle of Inclusion and Exclusion

Question

1000 households were surveyed.

275 households own a desktop computer,
455 households own a DVD player,
405 households own two cars,
145 households own a desktop computer and DVD player,
195 households own a DVD player and two cars,
110 househods own two cars and a desktop computer,
265 households do not own a desktop computer, do not
own a DVD player, do not own two cars.
Find the number of households that own:

c) a desktop computer, DVD player but do not
own two cars.

I have finished this problem but I am not sure if my answers are correct. Let $A = \text{desktop computers}$, $B = \text{DVD Players}$, and $C = \text{Two Cars}$.

Using the principle of inclusion/exclusion I found that $735$ households own something since $265$ households do not own $A,B,C$ and that $50$ households own all $A,B,C$.

c) a desktop computer, DVD player but do not own two cars. I let $y$ represent the shaded region.

$$|A \cap B|- |A \cap B \cap C| = y$$ $$145 - 50 = y$$ $$y=95$$

Is this the correct way to do this problem?

Answer 1

I think you did it correct...To have an intuition and confidence that you are correct, don't use any formula. Instead, mark all the areas in the venn diagram as 1,2,3,4...
$$|A \cap B|- |A \cap B \cap C| = (5 + 7) - 7 = 5$$.