19. Given two sets $A=\{1,2,3, \ldots, 15\}$ and $B=\{0,1\}$, find the number of mappings $f: A \rightarrow B$ with 1 being the image of at least two elements of $A$.
19. Ans: 32752

There are $2^{15}$ mappings from $A$ to $B$. There is only one mapping $f: A \rightarrow B$ with $f(i)=0$ for all $i \in A$; and there are 15 mappings $f: A \rightarrow B$ with $f(i)=0$ for all $i \in A \backslash\{k\}$ and $f(k)=1$, for $k=1,2, \ldots, 15$. Thus the answer is 32752 .