Prove that $$\sum\limits_{1\le x\le y\le t}\frac{2^{t-x}}{xy}=\sum\limits_{1\le x\le y\le t}\frac{\text C_t^0+\text C_t^1+\cdots+\text C_t^{y-1}}{xy},$$ where $t$ is a positive integer.
I think that we can find a combinatorial proof.
For example, $2^{t-x}$ seems to indicate the number of ways to choose any number of people among all $t$ people, excluding a certain group of $x$. The numerator on the right side may be explained by similar methods.
What's the most bothering is the denominator. It only makes sense in certain perspectives like probability.
From the algebraic side, I failed to use induction, because $t$ is appears everywhere.