133 of the cars are black, 401 have an automatic transmission, 57 have a sunroof, 116 are black and have an automatic transmission, 10 are black and have a sun roof, 42 have an automatic transmission and a sun roof, andare black, have an automatic transmission, and have a sun roof   a) How many cars are black and have an automatic transmission, but do not have a sun roof?  b) How many cars are not black, do not have an automatic transmission, and do not have a sun roof?

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There seems to be one number missing: the number of black cars with automatic transmission and a sun roof, so we'll call this number X. It's helpful to use a diagram (Venn diagram). Draw a large circle (ellipse or other enclosure) containing three interlocking circles. I'll use the word "circle" to mean any completely enclosed area. The three circles represent black cars (B), automatics (A), and cars fitted with a sunroof (S) respectively. The large circle represents all the cars (C), so those that are not black, not automatic and have no sunroof are represented by the interior of the large circle outside the other three circles.

We know there are 133 black cars, and 116 of these are automatic, so the remaining 17 must be manual transmission (non-automatic). We also know that 10 black cars have a sunroof, so 123 don't have a sunroof. And we know that X black automatics have a sunroof. 

Of the 401 automatics, 42 have sunroofs, so 359 automatics have no sunroof; and of the 57 cars fitted with a sunroof, 42 are automatics, so 15 are manual.

In the Venn diagram the intersection of the three sets are represented by the areas enclosed by two or three interlocking circles. I need a symbol to express intersection, normally represented by an inverted U symbol, which I don't have on my tablet, so I'll use ^, which is normally used to represent "to the power of". A^B means black automatics because it's the set of all black cars with automatic transmission, intersection of the black set B with the automatic set A. On the diagram it's the area enclosed by circles A and B. A^B^S is the area enclosed by the three interlocking circles and represents all black automatics with a sunroof. In all, the overlapping circles produce four enclosed areas: A^B, A^S, B^S and A^B^S. We can give values to these: 116, 42, 10 and X, respectively. Remember, we weren't given a value for X. We also have A=401, B=133, S=57.

We can represent "not" by underlining a set, so, for example, A represents non-automatic and B^A would mean black cars without automatic transmission (AT), so B^A=17 because 17 cars are black without AT. Similarly, B^S=123. So B^A^S=A^B^S=116-X. That is, a) there are 116-X black automatics without a sunroof.

There are 556-133=423 cars that are not black; 556-401=155 cars that are not automatic; and 556-57=499 cars without a sunroof.

b) The final part of the question is easier to work out by looking at the diagram. We need to work out the total number of cars within the interlocking circles. We can't simply add the numbers of cars in the three circles and subtract the sum from 556, because the circles interlock. Take A and B for example. They interlock enclosing 116 cars (black automatics), so we deduct 116 from A and add the result to B; or we deduct 116 from B and add the result to A: 285+133=418=17+401. 

If you look at the Venn diagram you'll see there are seven areas produced by the interlocking circles. Let's call the areas a, b, c, d, e, f and g, such that all automatics are given by a+d+g+e=401; black cars by b+d+g+f=133; and sunroofs by c+e+f+g=57. Diagrammatically, a is the set of automatics that are not black and have no sunroof; b the set of black cars that are neither automatic nor have a sunroof; and c the set of sunroofed cars that are neither black or automatic. There is an eighth area, h, outside of all the interlocking circles representing the residual cars that are not black, not automatic, and have no sunroof.

Area g contains X cars (black automatics with a sunroof), so g=X; d+g=116; f+g=10; e+g=42; a+b+c+d+e+f+g+h=556. So d=116-X; f=10-X; e=42-X; a=401-(116+42-X)=43+X; b=133-(116+10-X)=7+X; c=57-(42-X+10)=5+X; (43+X)+(7+X)+(5+X)+(116-X)+(42-X)+(10-X)+X+h=556; so h=556-(223+X)=333-X. This is the number of cars that are not black, not automatic and have no sunroof. Although we don't know the value of X, we know it must be less than 10 because area f contains a number of cars that can't, obviously, be negative. The wording of the question suggests that X is greater than 1, so the possible answers range from 324 to 331 (1<X<10), where X is, of course, an integer.

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