# Set Theory, Calendars, Clocks and Binomial Theorem

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## Sets and Unions

In a class of 345 students, the students who took English, Math and Science are equal in number. There are 30 students who took both English and Math, 26 who took both Math and Science, 28 who took Science and English and 14 who took all the 3 subjects.There are 43 students who didn’t take any of the subjects. Answer the following question according to the data given above.
How many students have taken English as a subject ?
1. 286
2. 124
3. 246
4. 108

Choice B. 124

## Detailed Solution

Let total no of students who took English be x
Then students who took math, science will also be x
Now let’s draw the Venn diagram

E U M U S = 345 – 43 (Neither of the subjects)
E U M U S = E + M + S – E ∩ M - E ∩ S - S ∩ M + E ∩ M ∩ S
=) 302 = 3x - 84 + 14
=) 302 + 84 – 14 = 3x
=) x = 372/3 = 124
Thus the total no of students who took English as a subject = 124
Consequently the Venn diagram becomes

Again,
The students who has taken only one subject = E U M U S - E ∩ M - E ∩ S - S ∩ M - E ∩ M ∩ S
= 302 - 16 - 14 - 14 - 12 = 246
The students who took English and Math but not science = only E + Only M + E ∩ M
= 80 + 82 + 16 = 178
Percent of students who took English and Math but not science = 178/302 x 100 = approx. 59 %
Correct Answer: 124, 246, 59%

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