JEE MAIN - Mathematics (2015 (Offline) - No. 20)
Let A and B be two sets containing four and
two elements respectively. Then, the number
of subsets of the set A $\times$ B , each having atleast
three elements are
219
256
275
510
Explanation
Given,
$$ \begin{aligned} &n(A)=4, n(B) =2 \\\\ &\Rightarrow n(A \times B) =8 \end{aligned} $$
Total number of subsets of set $(A \times B)=2^8$
Number of subsets of set $A \times B$ having no element (i.e. $\phi)=1$
Number of subsets of set $A \times B$ having one element $={ }^8 C_1$
Number of subsets of set $A \times B$ having two elements $={ }^8 C_2$
$\therefore$ Number of subsets having atleast three elements
$$ \begin{aligned} &=2^8-\left(1+{ }^8 C_1+{ }^8 C_2\right) \\\\ &=2^8-1-8-28 \\\\ &=2^8-37 \\\\ &=256-37=219 \end{aligned} $$
$$ \begin{aligned} &n(A)=4, n(B) =2 \\\\ &\Rightarrow n(A \times B) =8 \end{aligned} $$
Total number of subsets of set $(A \times B)=2^8$
Number of subsets of set $A \times B$ having no element (i.e. $\phi)=1$
Number of subsets of set $A \times B$ having one element $={ }^8 C_1$
Number of subsets of set $A \times B$ having two elements $={ }^8 C_2$
$\therefore$ Number of subsets having atleast three elements
$$ \begin{aligned} &=2^8-\left(1+{ }^8 C_1+{ }^8 C_2\right) \\\\ &=2^8-1-8-28 \\\\ &=2^8-37 \\\\ &=256-37=219 \end{aligned} $$
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