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Re: Each new car produced in a certain factory includes at least [#permalink]
computerbot, when we should subtract all three three times, not 2 times? IN the explanation, it says that items in all three categories are triple counted.
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Re: Each new car produced in a certain factory includes at least [#permalink]
computerbot wrote:
Here is a quicker solution.

We know that for three overlapping sets A, B, C:

Total = A + B + C - (sum of all 2s) - 2(sum of all threes)


Putting in the values from the prompt/question, we get;

50 = 25 + 13 + 30 - (sum of all 2s) - 2(sum of all threes)

50 = 68 - (sum of all 2s) - 2(sum of all threes)

We have to plug values from each option in the aforementioned equation to find the correct answers.




Option A: 2 cars with two options and 8 with all three options.

50 = 68 - (sum of all 2s) - 2(sum of all threes)
50 = 68 - (2) - 2(8)
50 = 68 - 18
50 = 50

Correct



Option B: 3 cars with two options and 7 with all three options.
50 = 68 - (sum of all 2s) - 2(sum of all threes)
50 = 68 - 3 - 2(7)
50 = 68 - 17
50 = 51

Incorrect

Option C: 4 cars with two options and 7 with all three options.
50 = 68 - (sum of all 2s) - 2(sum of all threes)
50 = 68 - 4 - 2(7)
50 = 68 - 18
50 = 50

Correct

Option D: 6 cars with two options and 6 with all three options.
50 = 68 - (sum of all 2s) - 2(sum of all threes)
50 = 68 - 6 - 2(6)
50 = 68 - 18
50 = 50

Correct

Option E: 7 cars with two options and 5 with all three options.
50 = 68 - (sum of all 2s) - 2(sum of all threes)
50 = 68 -7 - 2(5)
50 = 68 - 17
50 = 51

Incorrect

Option F: 8 cars with two options and 4 with all three options.
50 = 68 - (sum of all 2s) - 2(sum of all threes)
50 = 68 - 8 - 2(4)
50 = 68 - 16
50 = 52

Incorrect


why do we substract it with 2(sum of all threes)? do we require at least 1 that has all the 3?

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Re: Each new car produced in a certain factory includes at least [#permalink]
1
When we counted the sum of all 3, it was counted thrice while considering the sum of all 2. Hence, we need to subtract twice from thrice to get one.

ademeutia wrote:
why do we substract it with 2(sum of all threes)? do we require at least 1 that has all the 3?

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Re: Each new car produced in a certain factory includes at least [#permalink]
1
How I solved:
50=25+13+30-Both-(2*All)+Neither
Since we are told that neither is 0 in the stem (at least one of the modifications..), we can exclude that.
50=68-Both-(2*All)
18=B+2A
9-B/2=A
Since A must be an integer (there cannot be half of a completed car), we can eliminate options B and E, since B would be odd and thus give A as a non integer.
A is possible, if B=2, 9-1=8=A
C is possible, if B=4, 9-2=7=A
D is possible, if B=6, 9-3=6
E is not possible, as if B=8, 9-4=5=A. Note that if neither was not zero, then E could be possible - we would just need neither to be equal to 1 in that case.

Therefore A, C, and D are good.
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