Find the Value of n Using Sets and Counting Principle
Question:
Each set \( X_r \) contains 5 elements and each set \( Y_r \) contains 2 elements and
\[ \bigcup_{r=1}^{20}X_r=S=\bigcup_{r=1}^{n}Y_r \]If each element of \( S \) belongs to exactly 10 of the \( X_r \)’s and to exactly 4 of the \( Y_r \)’s, then find the value of \( n \).
Solution
Let the number of elements in \( S \) be \( m \).
Since each of the 20 sets \( X_r \) contains 5 elements,
\[ 20\times5=100 \]Also each element of \( S \) belongs to exactly 10 sets \( X_r \),
\[ 10m=100 \] \[ m=10 \]Now each set \( Y_r \) contains 2 elements.
Hence total number of element occurrences is
\[ 2n \]Since each element of \( S \) belongs to exactly 4 sets \( Y_r \),
\[ 4m=2n \] \[ 4\times10=2n \] \[ 40=2n \] \[ n=20 \]Hence, the required value of \( n \) is
\[ \boxed{20} \]