answer:This problem is a counting problem that involves classification. - First, there are students who can prove the problem using the synthetic method. There are 5 such students, so there are 5 ways to select a student from this group. - Second, there are students who can prove the problem using the analytical method. There are 3 such students, so there are 3 ways to select a student from this group. According to the principle of counting for different exclusive categories, the total number of ways to select a student, either from the group that can use the synthetic method or from the group that can use the analytical method, is the sum of these two possibilities. Therefore, the total number of selection methods is: text{Total selection methods} = 5 (text{using synthetic method}) + 3 (text{using analytical method}) = 8 So the total selection methods is boxed{8}, which corresponds to option A.

answer:We have f(x) = log_2(x^2 + 7) and a_n = f(n). To find the 5th term of {a_n}, we need to compute a_5. a_5 = f(5) = log_2(5^2 + 7) = log_2(25 + 7) = log_2(32). Since 2^5 = 32, we have log_2(32) = 5. Therefore, the 5th term of {a_n} is boxed{5}.

answer:To solve the problem, we need to understand what the absolute value function does. The absolute value of a number is the distance of that number from 0 on the number line, regardless of direction. Therefore, the absolute value of any number is always non-negative. Given the number -3, we apply the definition of absolute value: [ begin{align*} text{Absolute value of } -3 &= |-3| &= 3 end{align*} ] Thus, the correct answer is B: 3. So, we encapsulate the final answer as follows: boxed{B}.

answer:First, let's calculate the number of files Nancy has left after deleting the PPT files and 33 PDFs: - She deleted all 30 PPT files. - She deleted 33 out of 43 PDFs, leaving her with 43 - 33 = 10 PDFs. So, she has 30 Word files + 10 PDFs = 40 files remaining. Now, since Word files are twice as important as PDFs, we want to prioritize putting Word files in the folders. Each folder can contain a maximum of 7 files. We will fill the folders with Word files first and then add PDFs if there's space. Let's see how many full folders of Word files we can create: 30 Word files ÷ 7 files per folder = 4 full folders with 2 Word files left over. Now we have 4 full folders of Word files and 2 Word files that need a folder. We also have 10 PDFs to allocate. Since we want to prioritize Word files, we will create a new folder for the remaining 2 Word files and then fill the rest of that folder with PDFs. So, we add 5 PDFs to the folder with the 2 Word files (since 2 Word + 5 PDF = 7 files, the maximum per folder). This leaves us with 10 - 5 = 5 PDFs remaining. These 5 PDFs will go into a new folder, but since we can't fill it completely (as we only have 5 PDFs and the folder can hold 7), this will be a partially filled folder. In total, Nancy would have: - 4 folders filled with 7 Word files each. - 1 folder with 2 Word files and 5 PDFs. - 1 folder with 5 PDFs. So, Nancy would have boxed{6} folders in total.