Bunuel wrote:
Each new car produced in a certain factory includes at least one of the following three options: Bluetooth, navigation system, and sunroof. Of the cars that came off the production line yesterday, 25 had Bluetooth, 13 had a navigation system, and 30 had a sunroof. Yesterday, 50 cars were completed. Which of the following could be true about the cars that came off the production line yesterday?
I There were 2 cars that had exactly two of the options and there were 8 cars that had all three options.
II. There were 4 cars that had exactly two of the options and there were 7 cars that had all three options.
III. There were 6 cars that had exactly two of the options and there were 6 cars that had all three options.
A. I only
B. II only
C. I and II only
D. II and III only
E. I, II, and III
Solution:We can use the formula:
Total = n(A) + n(B) + n(C) - n(Exactly two sets) - 2 * n(All three sets) + n(No set)
Since n(No set) = 0, we have:
Total = n(A) + n(B) + n(C) - n(Exactly two sets) - 2 * n(All three sets)
Substituting the numbers, we have:
50 = 25 + 13 + 30 - n(Exactly two sets) - 2 * n(All three sets)
50 = 68 - n(Exactly two sets) - 2 * n(All three sets)
n(Exactly two sets) + 2 * n(All three sets) = 18 → Eq. 1
Now, let’s analyze each statement.
I. There were 2 cars that had exactly two of the options and there were 8 cars that had all three options.
n(Exactly two sets) = 2 and n(All three sets) = 8
Let’s substitute this into Eq. 1 and see if it works:
2 + 2 * 8 = 18 ?
18 = 18 ? → Yes
Statement I could be true.
II. There were 4 cars that had exactly two of the options and there were 7 cars that had all three options.
n(Exactly two sets) = 4 and n(All three sets) = 7
Let’s substitute this into Eq. 1 and see if it works:
4 + 2 * 7 = 18 ?
18 = 18 ? → Yes
Statement II could be true.
II. There were 6 cars that had exactly two of the options and there were 6 cars that had all three options.
n(Exactly two sets) = 6 and n(All three sets) = 6
Let’s substitute this into Eq. 1 and see if it works:
6 + 2 * 6 = 18 ?
18 = 18 ? → Yes
Statement III could be true.
Answer: E
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