Question

Let X be a set of $5$ elements. The number d of ordered pairs (A, B) of subsets of X such that $A\ne \Phi ,B\ne \Phi ,A\cap B=\Phi$ satisfies.

• A. $50\le d\le 100$
• B. $101\le d\le 150$
• C. $151\le d\le 200$
• D. $201\le d$

Answer 1

Answer: (C)

$151\le d\le 200$

Total no. of ordered pairs is

${=C}_2^5\times 2!+C_3^5(\frac{3!}{1!\times 2!}\times 2!)+C_4^5(\frac{4!}{1!\times 3!}\times 2!+\frac{4!}{2!\times 2!}\times \frac{2!}{2!})+C_5^5(\frac{5!}{1!\times 4!}\times 2!+\frac{5!}{2!\times 3!}\times 2!)=10\left(2\right)+10\left(6\right)+5(8+6)(10+20)=180$

So option $C$ is correct