How many sets in the power set containing a given integer? Let $\mathcal{J}\equiv \{1,...,J\}$ and let $\mathcal{C}$ be the power set of $\mathcal{J}$ (with cardinality $2^{J}$).
Question: take any $j\in \mathcal{J}$. How many elements of $\mathcal{C}$ (sets) contain $j$?
For example: if $J=3$ and $j=1$, then $\mathcal{C}\equiv \Big\{\emptyset, \{1,2,3\}, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}\Big\}$ and there are $4$ sets in $\mathcal{C}$ containing $1$. 

Can you find a bijective function that maps each element of the power set of $\mathcal{J}-\{j\}$ to the set that you want to count?
