Placement Exams

Probability Level 2

To be accepted to the Science Academy each of 100 students has to pass a written placement exams in Mathematics, Physics, Chemistry, Biology and Geology. The following is the results:

90 students passed the Mathematics exam
89 students passed the Physics exam
86 students passed the Chemistry exam
92 students passed the Biology exam
88 students passed the Geology exam

What is the minimum number of students that passed all five exams?


The answer is 45.

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Paul Fournier by Paul Fournier , 70, Canada

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4 solutions

Yash Jain
Apr 29, 2016

First calculate the total number of students failed in each subject:

Mathematics: 10 Physics: 11 Chemistry: 14 Biology: 8 Geology: 12

Now we need to calculate the minimum students who cleared all five exams which means we need to subtract the maximum failed students from 100. So instead of making combinations adding all the number of students failed in the examination will give us the maximum number of students failed.

Minimum number of students who cleared the exam ⇒ 100 − ( 10 + 11 + 14 + 8 + 12 ) = 45 \Rightarrow 100 - (10+11+14+8+12) = \boxed{45}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

There are 55 failed results. To have the minimum number of students that passed all 5, we must have the most number of students which didn't pass all 5. This means the students that didn't pass must each fail 1 exam so the 55 failed results must be shared amongst 55 students to maximize the number of students that didn't pass. This is opposed to having only 11 students who failed all 5, which will mean the maximum number (89). Therefore, 100 - 55 is 45.

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But the max could not be 89, since only 86 passed Chemistry exam.

Saya Suka - 2 years ago
Shantanu Sawale
Apr 30, 2016

If you calculate the total number of failed students for every subjeact and add them considering all different. You will end up with 45 as your answer

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Jamie Simonsen
Apr 23, 2016

Its just how ever many failed each test e.g 10, 11, 14 ..... These are all added and subtracted from 100

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I agree that the worst case scenario is when the 10 that failed Math, the 11 that failed Physics, the 14 that failed Chemistry, the 8 that failed Biology and the 12 that failed Geology are all different students. Thus 100 minus those who failed exactly one exam. 100-55=45

Paul Fournier - 5 years, 1 month ago

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