Points: 0 of 1 Save Use the Venn diagram and the given conditions to determine the number of elements in each region, or explain why the conditions are impossible to meet. \[ n(U)=43, n(A)=24, n(B)=20, n(C)=18, n(A \cap B)=10, n(A \cap C)=11, n(B \cap C)=7 \text {, } \] \[ n(A \cap B \cap C)=5 \] First decide whether or not the conditions are possible to meet. Select the correct choice below and fill in the answer box(es) within your choice. A. The number of elements in regions I, II, III, IV, V, VI, VII, VIII are $\square, \square, \square, \square, \square, \square, \square, \square$, respectively. B. It is impossible to meet the conditions because there are only $\square$ elements in set A but there are $\square$ elements in set A that are also in set B or C. A similar problem exists for set C. (Simplify your answers.) C. It is impossible to meet the conditions because there are only $\square$ elements in set B but there are $\square$ elements in set B that are also in set A or C. A similar problem exists for set C. (Simplify your answers.)
Solution
Step 1 ：Let's denote the regions as follows: I: Only in A, II: Only in B, III: Only in C, IV: In A and B, but not C, V: In A and C, but not B, VI: In B and C, but not A, VII: In A, B, and C, VIII: In U, but not in A, B, or C
Step 2 ：We know that \(n(A \cap B \cap C) = 5\), so VII = 5
Step 3 ：We also know that \(n(A \cap B) = 10\), but this includes the elements that are also in C. So, IV = \(n(A \cap B) - VII = 10 - 5 = 5\)
Step 4 ：Similarly, V = \(n(A \cap C) - VII = 11 - 5 = 6\), and VI = \(n(B \cap C) - VII = 7 - 5 = 2\)
Step 5 ：Now, we can find the number of elements that are only in A, B, or C. I = \(n(A) - IV - V - VII = 24 - 5 - 6 - 5 = 8\). II = \(n(B) - IV - VI - VII = 20 - 5 - 2 - 5 = 8\). III = \(n(C) - V - VI - VII = 18 - 6 - 2 - 5 = 5\)
Step 6 ：Finally, we can find the number of elements that are in U but not in A, B, or C. VIII = \(n(U) - I - II - III - IV - V - VI - VII = 43 - 8 - 8 - 5 - 5 - 6 - 2 - 5 = 4\)
Step 7 ：So, the number of elements in regions I, II, III, IV, V, VI, VII, VIII are 8, 8, 5, 5, 6, 2, 5, 4, respectively. Therefore, the answer is \(\boxed{A}\)
