1. A commercial lock with 10 buttons, which can be opened by pressing the correct five numbers, regardless of the order. The figure below is an example using $\{1,2,3,6,9\}$ as its combination. If these locks are reprogrammed to allow combinations of one to nine digits, how many additional combinations (i.e., not using five digits) are possible?
1. ( 770$)$

There are $2^{10}$ ways to choose from ten buttons. From this, we need to subtract the case where all 10 digits are selected, the case where no digits are selected, and the case where exactly five digits are selected. Therefore, the total number is
$$
2^{10}-1-1-C_{10}^{5}=1024-2-252=770 .
$$