+12 Cora has exactly four wooden toy blocks with digits on them. Two of the blocks have the digit 0, 1 of the blocks has the digit 1, and one of the blocks has the digit 2. How many distinct positive integers can she form using one or more of the blocks, without using each block more than once?
We can proceed by casework on the number of digits used. Case 1: Cora uses 1 digit. There are two cases: (i) she uses the digit 0, in which case she can form the numbers 0, 1, 2; (ii) she uses the digit 1, in which case she can form the numbers 1 and 2. Case 2: Cora uses 2 digits. There are two cases: (i) she uses the digits 0 and 1, in which case she can form the numbers 01, 02, 12; (ii) she uses the digits 1 and 2, in which case she can form the numbers 12. Case 3: Cora uses 3 digits. She can only form the number 120. Case 4: Cora uses 4 digits. She can only form the number 120. Thus, the total number of distinct positive integers Cora can form is 2+2+3+1=8.
