Choose the correct answers from the given four option: Suppose A_… - Vidyadip

MCQ

Choose the correct answers from the given four option: Suppose $A_1, A_2, ..., A_{30}$ are thirty sets each having $5$ elements and $B_1, B_2, ..., Bn$ 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 $A_i$ ’s and exactly $9$ of the $B, 'S.$ then $n$ is equal to.

A
$15$
B
$3$
✓
$45$
D
$35$

✓

Answer

Correct option: C.

$45$

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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