5. If a positive integer with at least two digits is such that all digits except the leftmost one are smaller than the digit to their left, then such a positive integer is called a "good number". Then, the total number of all such good numbers is ( ).
(A) 1013
(B) 1011
(C) 1010
(D) 1001
5.A.

First, consider the number of good numbers with $k(2 \leqslant k \leqslant 10)$ digits is $\mathrm{C}_{10}^{k}$. Therefore, there are $\sum_{k=2}^{10} \mathrm{C}_{10}^{k}=\sum_{k=0}^{10} \mathrm{C}_{10}^{k}-\mathrm{C}_{10}^{1}-\mathrm{C}_{10}^{0}=1013$ in total.