X has 10 elements, P(X) its power set.

Question

$X$ has 10 elements, $P(X)$ its power set. $A,B$ picked at random from $P(X)$ with replacement. Find the probability $A,B$ have equal numbers of elements.

Accepted Answer

$$P = \frac{\sum_{k=0}^{10}({}^{10}C_k)^2}{2^{10}\times2^{10}} = \frac{{}^{20}C_{10}}{2^{20}}$$

(using the Vandermonde identity $\sum({}^{10}C_k)^2={}^{20}C_{10}$).

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