To solve for the number of elements in the intersection of two sets [tex]\( n(A \cap B) \)[/tex], we can use the principle of inclusion and exclusion for sets. The principle of inclusion and exclusion for two sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] states:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
We are given:
- [tex]\( n(U) = 100 \)[/tex] (the total number of elements in the universal set, although we don't need this directly for our calculation)
- [tex]\( n(A) = 75 \)[/tex] (the number of elements in set [tex]\( A \)[/tex])
- [tex]\( n(B) = 40 \)[/tex] (the number of elements in set [tex]\( B \)[/tex])
- [tex]\( n(A \cup B) = 80 \)[/tex] (the number of elements in the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex])
To find [tex]\( n(A \cap B) \)[/tex], we rearrange the formula to solve for the intersection:
[tex]\[ n(A \cap B) = n(A) + n(B) - n(A \cup B) \][/tex]
Substituting the given values into the equation:
[tex]\[ n(A \cap B) = 75 + 40 - 80 \][/tex]
[tex]\[ n(A \cap B) = 115 - 80 \][/tex]
[tex]\[ n(A \cap B) = 35 \][/tex]
Therefore, the number of elements in the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( 35 \)[/tex].
[tex]\[ \boxed{35} \][/tex]
