4. Find the number of mathematics students at a college taking at least one of t

Question

4. Find the number of mathematics students at a college taking at least one of the languages French, German, and Russian, given the following data: 65 study French 45 study German 42 study Russian 20 study French and German 25 study French and Russian 15 Study German and Russian 8 study all these languages 5. How many positive integers between 1000 and 9999 inclusive are: a) divisible by 9? b) are even? c) have distinct digits? d) are not divisible by 3? e) are divisible by 5 or 7? f) are not divisible by either 5 or 7? g) are divisible by 5 but not by 7? h) are divisible by 5 and 7?

Explanation / Answer

The answer is 100. Here's how we get there. First of all, obviously the number of people who study all three languages is 8. Now let's look at people who study exactly two of the languages: 20 study French and German, but of those, 8 study Russian as well, so a total of 12 study only French and German. 25 study French and Russian, but of those, 8 study German as well, so a total of 17 study only French and Russian. Similarly, a total of 7 study only German and Russian. In total, 36 people study exactly two languages. Now let's look at people who study exactly one language: 65 people study French in total. Of these, 12 study exactly the two languages French and German (from above), 17 study exactly the two languages French and Russian (from above) and 8 study all three languages. The remaining 28 people study only French. 45 people study German in total. Of these, 12 study exactly the two languages French and German (from above), 7 people study exactly the two languages German and Russian (from above) and 8 study all three languages. The remaining 18 people study only German. Similarly, 10 people study only Russian. In total, 56 people study exactly one language. Now, a person who takes at least one of the languages must either take exactly one language, exactly two languages or exactly three languages. Therefore the answer is 56+36+8=100. In full, the calculation is: 8+((20-8)+(25-8)+(15-8))+((65 -(20-8)-(25-8)-8)+(45-(20-8) -(15-8)-8)+(42-(25-8)-(15-8) -8)) =8+(12+17+7)+((65-12-17-8) +(45-12-7-8)+(42-17-7-8)) =8+36+(28+18+10) =8+36+56 =100

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