Use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. \[ \begin{array}{l} n(A)=20 \\ n(B)=60 \\ n(A \cap B)=15 \\ n(U)=200 \end{array} \] (A) Find $n(A \cap B)$. \[ n(A \cap B)=15 \] (B) Find $n\left(A \cap B^{\prime}\right)$. \[ n\left(A \cap B^{\prime}\right)=5^{\prime} \] (C) Find $n\left(A^{\prime} \cap B\right)$. \[ n\left(A^{\prime} \cap B\right)=45 \] (D) Find $n\left(A^{\prime} \cap B^{\prime}\right)$. \[ n\left(A^{\prime} \cap B^{\prime}\right)=\square \]

Step 1 ：Given that the number of elements in set A, denoted as \(n(A)\), is 20.

Step 2 ：Given that the number of elements in set B, denoted as \(n(B)\), is 60.

Step 3 ：Given that the number of elements in the intersection of set A and set B, denoted as \(n(A \cap B)\), is 15.

Step 4 ：Given that the number of elements in the universal set U, denoted as \(n(U)\), is 200.

Step 5 ：The number of elements in the subset that is not in A and not in B, denoted as \(n(A^{\prime} \cap B^{\prime})\), can be found by subtracting the number of elements in A, B and the intersection of A and B from the total number of elements in the universal set U.

Step 6 ：So, \(n(A^{\prime} \cap B^{\prime}) = n(U) - n(A) - n(B) + n(A \cap B)\).

Step 7 ：Substitute the given values into the equation, we get \(n(A^{\prime} \cap B^{\prime}) = 200 - 20 - 60 + 15\).

Step 8 ：Simplify the equation, we get \(n(A^{\prime} \cap B^{\prime}) = 135\).

Step 9 ：Final Answer: The number of elements in the subset that is not in A and not in B is \(\boxed{135}\).