12. The number 3 can be expressed as the sum of one or more positive integers in four different ways:
$$
3 ; \quad 1+2 ; \quad 2+1 ; \quad 1+1+1 .
$$

In how many ways can the number 5 be so expressed?
A 8
B 10
C 12
D 14
E 16
Solution: $\mathbf{E}$
By systematically listing all the possibilities, we see that the number 5 may be written as a sum of one or more positive integers in the following 16 ways:
$$
\begin{array}{c}
5 ; 1+4 ; 4+1 ; 2+3 ; 3+2 ; 3+1+1 ; 1+3+1 ; 1+1+3 ; 1+2+2 ; 2+1+2 ; \\
2+2+1 ; 1+1+1+2 ; 1+1+2+1 ; 1+2+1+1 ; 2+1+1+1 ; 1+1+1+1+1
\end{array}
$$