• # question_answer 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) $201\le d$

[c]  We have, $n(X)=5$ A and B are subset of X Such that $A\ne \phi ,$$B\ne \phi A\cap B=\phi$ Total number of order pair (A, B) is ${}^{5}{{C}_{2}}\times 2!+{}^{5}{{C}_{3}}\left( \frac{3!}{2!}\times 2! \right)+{}^{5}{{C}_{4}}\left( \frac{4!}{3!}2!+\frac{4!}{2!2!} \right)$$+{}^{5}{{C}_{5}}\left( \frac{5!}{4!}2!+\frac{5!}{2!3!}\times 2! \right)$ $=10\,(2)+10\,(6)+5\,(8+6)+(10+20)$ $=20+60+70+30=180$ $\therefore$$d\in [151,200]$      $\Rightarrow$$151\le d\le 200$