Suppose A1, A2, ..., A30 are thirty sets each having 5 elements and B1, B2, ..., Bn are n sets each with 3 elements, let $\bigcup\limits_{\text{i}=1}^{30}\text{A}_\text{i}=\bigcup\limits_{\text{j}=1}^\text{n}\text{B}_\text{j}=\text{S}$ and each element of S belongs to exactly 10 of the Ai ’s and exactly 9 of the B, 'S. then n is equal to.
Solution:
Number of elements in $\text{A}_1\cup\text{A}_2\cup\text{A}_3\ ..... \cup \text{A}_{30}=30\times5=150$ (When repetition is not allowed)
But each element is repeated 10 times
$\therefore \text{n(S)}=\frac{30\times5}{10}=\frac{150}{10}=15\ .....\text{(i)}$
Number of elements in $\text{B}_1\cup\text{B}_2\cup\text{B}_3\ ...... \text{B}_\text{n}=3\text{n}$ (when repetitiom is not allowed)
But each element is repeated 09 times
$\therefore \text{n(S)}=\frac{3\text{n}}{9}=\frac{\text{n}}{3}\ .....\text{(ii)}$
From (i) and (ii) we get
$\frac{\text{n}}{3}=15\Rightarrow \text{n}=15\times3=45$
Hence, the corrrect option is (c).
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