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 \(n\) is
(a) 10
(b) 20
(c) 100
(d) 50
Solution
Total element occurrences in all \(X_r\)’s:
\[ 20\times5=100 \]
If \(S\) has \(k\) elements and each element occurs in 10 sets,
\[ 10k=100 \]
\[ k=10 \]
Now total element occurrences in all \(Y_r\)’s:
\[ 2n \]
Each element of \(S\) occurs in 4 sets,
\[ 4\times10=2n \]
\[ 40=2n \]
\[ n=20 \]
Answer
\[ \boxed{20} \]
Correct option: (b)