Set Theory, Calendars, Clocks and Binomial Theorem

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Set Theory: Min and Max % of people

    95% of the students in a class have taken Marketing, 80% have chosen Finance, 84% have chosen operations (ops), and 90% have chosen Human Resources (HR). What is the maximum and minimum percentage of people who have chosen all of the four?
    1. 80% and 56%
    2. 95% and 53%
    3. 80% and 49%
    4. 80% and 51%


  • Correct Answer
    Choice C. 80% and 49%

Explanatory Answer

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Detailed Solution

If you get the Venn Diagram right, you are halfway through !!

Finding the maximum percentage is easy. If , then the percentage of people who have taken all 4 should be 80% and this is the maximum value it can take.

For the value to be minimum, the numbers should be as far ‘apart´ as possible. Let us do this iteratively. First, let us take Marketing and Finance, and see if we can find the minimum percentage of students who should have taken both.

When M and F, are as far ‘apart´ as possible, would be minimum. And the minimum value would be 80% + 95% – 100% = 75%.

Now, let us start with this and add Ops to the mix.

Now, the minimum value of would be when these are as far ‘apart’ as possible. And the minimum value would be 75% + 84% – 100% = 59%.

Adding HR also to the mix, we get -

The minimum possible value of = 59% + 90% – 100% = 49%.

As a formula, the minimum value is 100% – (100% – 95%) – (100% – 80%) – (100% – 84%) – (100% – 90%).

= 100% – 5% – 20% – 16% – 10% = 100% – 51% = 49%.

Again, do not go with just the formula. See the trial and error iteration. The trial and error iteration is far more useful than knowing the formula for a template.

Answer choice (c)

Correct Answer: 80% and 49%.

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