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Sets — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSets1 Mark
Question
Suppose A1, A2, ..., A30  are thirty sets each having 5 elements and B1, B2, ..., Bn are n sets each with 3 elements. Let $\bigcup\limits^{30}_\text{i = 1}\text{A}_\text{i}=\bigcup\limits^{\text{n}}_\text{j = 1}\text{B}_\text{j}=\text{S}$  and each element of S belong to exactly 10 of the Ai's and exactly 9 of the Bj's, then n is equal to:
  1. 15
  2. 3
  3. 45
  4. 35.
✓

Answer

  1. 45.

Solution:

It is given that each set $\text{A}_\text{j}(1\leq\text{i}\leq30)$ contains 5 elements and $\bigcup\limits^{30}_\text{i = 1}\text{A}_\text{i}=\text{S}.$

$\therefore\text{n(S)}=30\times5=150$

But, it is given that each element of S belong to exactly 10 of the Ai's.

$\therefore$ Number of distinct elements in $\text{S}=\frac{150}{10}=15......(1)$

It is also given that each set $\text{B}_\text{j}(1\leq\text{j}\leq\text{n})$ contains 3 elements and $\bigcup\limits^{\text{n}}_\text{j = 1}\text{B}_\text{j}=\text{S}.$

$\therefore\text{ n(S)}=\text{n}\times3=\text{3n}$

Also, each element of S belong to eactly 9 of Bj's.

$\therefore$ Number of distinct elements in $\text{S}=\frac{\text{3n}}{9}......(2)$

From (1) and (2), we have

$\frac{\text{3n}}{9}=15$

$\Rightarrow\text{n} = 45.$

Hence, the correct answer is option (c).

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