Let's Unite Them!

Discrete Mathematics Level pending

Let us say \(A_1,A_2,A_3,\ldots ,A_{30}\) are thirty sets containing 6 elements each. While \(B_1,B_2,B_3,\ldots ,B_n\) are \(n\) sets containing 3 elements each. Now consider the following;

\[\large\displaystyle \mathop{\bigcup}_{k=1}^{30}A_k=\mu=\displaystyle \mathop{\bigcup}_{k=1}^{n}B_k\]

Such that, each element of \(\mu\) belongs to exactly 10 elements of \(A_k\)'s and exactly 9 elements of \(B_k\)'s, then find the value of \(n\).

Notation: The symbol \(\cup\) denotes set union, and \[\large\displaystyle\mathop{\bigcup}_{k=1}^{m}X_k=X_1\cup X_2\cup X_3\cup\dots\cup X_m.\]


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