42. Of the seniors, some study French (F), some study Spanish (S), and some both. If 45 study at least one of the two languages, 20 study Spanish, and 12 study both, how many study French?
Answer: C
37 students study French.
To find out how many students study French, we can use the principle of inclusion-exclusion. Given that 45 students study at least one of the two languages, 20 study Spanish, and 12 study both, we can calculate the number of students studying French.
A) 25
This option is incorrect. If 25 students studied French, that would imply a total of only 33 students studying at least one language when combined with the number studying Spanish and both, which does not satisfy the total of 45 students studying at least one language.
B) 33
This option is incorrect. If 33 students studied French, this would mean that 33 (French) + 20 (Spanish) - 12 (both) equals 41 students studying at least one language, which is less than the given total of 45 students.
C) 37
This option is correct. By calculating the number of French students using the equation: Total = (French + Spanish - Both), we have 45 = (F + 20 - 12). Rearranging gives us F = 37, indicating that 37 students study French.
D) 41
This option is incorrect. If 41 students studied French, then the total would be 41 (French) + 20 (Spanish) - 12 (both), resulting in 49 students studying at least one language, which exceeds the provided total of 45.
Conclusion
The correct answer of 37 students studying French is derived from the proper application of the inclusion-exclusion principle, confirming that the calculations align with the total number of students studying at least one language. All other options fail to meet the criteria set by the problem, either exceeding the total or falling short.