SOLUTION: Among 18 students in a room, 7 study mathematics, 10 study science, and 10 study computer programming. Also, 3 study mathematics and science, 4 study mathematics and computer pro

Question 1158213: Among 18 students in a room, 7 study mathematics, 10 study science, and 10 study computer
programming. Also, 3 study mathematics and science, 4 study mathematics and computer
programming, and 5 study science and computer programming. We know that 1 student studies all three
subjects. How many of these students study none of the three subjects?

Answer by ikleyn(52780) (Show Source): You can put this solution on YOUR website!
.
Among 18 students in a room, 7 study mathematics, 10 study science, and 10 study computer
programming. Also, 3 study mathematics and science, 4 study mathematics and computer
programming, and 5 study science and computer programming. We know that 1 student studies all three
subjects. How many of these students study none of the three subjects?
~~~~~~~~~~~~~~~

Let M be the set of those who study (at least) Math     ( n(M) = 7 )

Let S be the set of those who study (at least) Science  ( n(S) = 10 )

Let C be the set of those who study (at least) Computer ( n(C) = 10 )


Let MS, MC and SC be the corresponding intersetion sets (those who study at least two subjects)

Let MSC be the intersection set (M & S & C).


There is a REMARCABLE formula in elementary set theory

    n(M U S U C) = n(M) + n(S) + n(C) - n(MS) - n(MC) - n(SC) + n(MSC).


It allows calculating the number of elements in the UNION of separate sub-sets.


In our case, this formula gives the value

    n(M U S U C) = 7 + 10 + 10 - 3 - 4 - 5 + 1 = 16.


Thus 16 students study at least one of listed subjects.


Hence, the rest 18-16 = 2 in the room study none of the three subjects.    ANSWER

Solved.

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To learn more about this formula, see the lessons
    - Counting elements in sub-sets of a given finite set
    - Advanced problems on counting elements in sub-sets of a given finite set
    - Challenging problems on counting elements in subsets of a given finite set
in this site.

You will find there a variety of other similar solved problems.

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