Question

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 at least three elements is............

• A. $275$
• B. $510$
• C. $219$
• D. $256$

Answer 1

The correct option is C $219$

Set A has $4$ elements

Set B has $2$ elements

Number of elements in set $(A\times B)=4\times 2=8$

Total number of subsets of $(A\times B)=2^8=256$

Number of subsets having $0$ elements ${=}^8{C}_0=1$

Number of subsets having $1$ elements each ${=}^8{C}_1=8$

Number of subsets having $2$ elements each ${=}^8{C}_2=\frac{8!}{2!6!}=28$

Number of subsets having atleast $3$ elements $=256-1-8-28$

$=256-37$

$=219$

Hence, the answer is $219.$