  Question

Let $$X$$ be a set containing $$10$$ elements and $$P(X)$$ be its power set. If $$A$$ and $$B$$ are picked up at random from $$P(X)$$, with replacement, then the probability that $$A$$ and $$B$$ have equal number of elements, is :

A
20C10210  B
(2101)220  C
20C10220  D
(2101)210  Solution

The correct option is C $$\displaystyle \frac{^{20}{C}_{10}}{2^{20}}$$$$X$$ is a set containing $$10$$ elements.Then power set $$P(X)$$ contains $$2^{10}$$ elements.$$A$$ and $$B$$ are picked at random from $$P(X)$$ with replacement.Total no. of outcomes are $$2^{10}.$$no. of ways in which $$A$$ and $$B$$ have equal no of elements is $$\displaystyle^{10}C_0^{10}C_0+^{10}C_1^{10}C_1+^{10}C_2^{10}C_2+....+^{10}C_{10}^{10}C_{10}$$$$\therefore$$ Probability $$=\displaystyle\frac^{10}C_0^{10}C_0+^{10}C_1^{10}C_1+^{10}C_2^{10}C_2+....+^{10}C_{10}^{10}C_{10}}{2^{20}$$                           $$=\displaystyle\frac{^{20}C_{10}}{2^{20}}$$Maths

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