<PRE><B>Question 1158213</B>
.
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?
~~~~~~~~~~~~~~~


&lt;pre&gt;
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 &amp; S &amp; 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.    &lt;U&gt;ANSWER&lt;/U&gt;
&lt;/pre&gt;

Solved.


--------------


To learn more about this formula, see the lessons

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=http://www.algebra.com/algebra/homework/word/misc/Counting-elements-in-sub-sets-of-a-given-finite-set.lesson&gt;Counting elements in sub-sets of a given finite set&lt;/A&gt;

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=https://www.algebra.com/algebra/homework/word/misc/Advanced-probs-counting-elements-in-sub-sets-of-a-given-finite-set.lesson&gt;Advanced problems on counting elements in sub-sets of a given finite set&lt;/A&gt;

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=https://www.algebra.com/algebra/homework/word/misc/Challenging-problems-on-counting-elements-in-subsets-of-a-given-finite-set.lesson&gt;Challenging problems on counting elements in subsets of a given finite set&lt;/A&gt; 

in this site.


You will find there a variety of other similar solved problems.


</PRE>