n Q ∩ R there are 9 elements.
In Q ∩ S there are 4 elements.
In R ∩ S there are 11 elements.
If (Q ∩ R) ∪ (Q ∩ S) ∪ (R ∩ S) includes 18 elements, how many elements are in Q ∩ R ∩ S?
To find the number of elements in \(Q \cap R \cap S\), we can use the principle of inclusion-exclusion.
From the given information, we know the following:
\(Q \cap R\) has 9 elements, \(Q \cap S\) has 4 elements, and \(R \cap S\) has 11 elements.
To find the number of elements in \(Q \cap R \cap S\), we can use the formula:
\[|Q \cap R \cap S| = |Q| + |R| + |S| - |Q \cup R| - |Q \cup S| - |R \cup S| + |Q \cup R \cup S|\]
Substituting the given information:
\[|Q \cap R \cap S| = 9 + 4 + 11 - |Q \cup R| - |Q \cup S| - |R \cup S| + 18\]
Now, we need the values of \(|Q \cup R|\), \(|Q \cup S|\), and \(|R \cup S|\).
Using the inclusion-exclusion principle:
\[|Q \cup R| = |Q| + |R| - |Q \cap R|\]
\[|Q \cup S| = |Q| + |S| - |Q \cap S|\]
\[|R \cup S| = |R| + |S| - |R \cap S|\]
Substituting the given information:
\[|Q \cup R| = |Q| + |R| - |Q \cap R| = 9 + |R| - 9 = |R|\]
\[|Q \cup S| = |Q| + |S| - |Q \cap S| = 9 + 4 - 4 = 9\]
\[|R \cup S| = |R| + |S| - |R \cap S| = |R| + 11 - 11 = |R|\]
Now we can substitute these values back into the original equation:
\[|Q \cap R \cap S| = 9 + 4 + 11 - |Q \cup R| - |Q \cup S| - |R \cup S| + 18\]
\[|Q \cap R \cap S| = 9 + 4 + 11 - |R| - 9 - |R| + 18\]
\[|Q \cap R \cap S| = 33 - 2|R|\]
Since we don't have the specific value of \(|R|\), we can't determine the exact number of elements in \(Q \cap R \cap S\) without further information.