# Set Theory, Calendars, Clocks and Binomial Theorem

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## Sets and Unions

In class of 260 students, each student needs to choose between the three extra subject (i.e IT, Hindi and Sanskrit) offered along with the course. The students that choose each of these subjects are 160, 130, 110. The number of students who choose more than one of the three is 40% more than the number of students who choose all the three subjects If there are no students who choose none of the 3 subjects, how many students study all the three subjects?
1. 40
2. 50
3. 80
4. 100

Choice B.50

## Detailed Solution

In this question there are 260 students who are counted 160 + 130 + 110 = 380 times
This means there is an extra count of 380 – 260 = 120 students
Now, let the no. of students who choose all the three Subject be ‘n’
Then as per the question, the no. of students who choose more than 1 subject = 1.4 n
Thus, the no of students who study exactly 2 subjects = 1.4 n – n = 0.4 n
Extra count can occur from exactly 2 areas i.e from the ‘exactly two areas’ or ‘all three area’. We also know that a student placed in in all the three area will be counted 3 times, thus the extra count being 2 while in the case of exactly two students extra count is 1.
Thus the extra count from n students choose 3 subjects would be = 2 x n = 2n
And that of 0.4 student choose exactly 2 subjects = 1 x 0.4 n = 0.4n
Hence extra count = 120 = 2n + 0.4n
=) 120 = 2.4 n
=) $\frac{120}{2.4}$ = n
=) 50 = n
Hence the no of students who choose all the three subjects = 50

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