Suppose $\mathrm{A}_1, \mathrm{~A}_2, \ldots \mathrm{~A}_{30}$ are thirty sets each having 5 element

Question

Suppose $\mathrm{A}_1, \mathrm{~A}_2, \ldots \mathrm{~A}_{30}$ are thirty sets each having 5 elements and $\mathrm{B}_1, \mathrm{~B}_2, \ldots \mathrm{~B}_{\mathrm{n}}$ are n sets each having 3 elements. Let $\quad \bigcup_{i=1}^{30} A_i=\bigcup_{j=1}^n B_j=S$. If each element of S belongs to exactly 10 of the $\mathrm{A}_{\mathrm{i} 	ext { 's }}$ and exactly 9 of the $\mathrm{B}_{\mathrm{j} 	ext { 's }}$ then n is equal to

Professor PC Thomas Classes · Answer

Let S contain k elements. Each element of $S$ is contained 10 of Ai's and each $A_i$ has 5 elements \ $ 	herefore 10 \mathrm{k}=30 	imes 5 \cdot \mathrm{k}=15 $ Each element of S is contained in $9 \mathrm{~B}_{\mathrm{j}}$ 's and each $B_j$ has 3 elements $ \ \begin{aligned} & 	herefore 9 	imes 15=3 n \ & 	herefore n=45 \end{aligned} $