How many strings of 12 decimal digits begin with "248" or consist only of even digits?

We need to find the number of strings in each of the two cases and then subtract the number that are in both. That is Event A or Event B = Event A + Event B - Event A and B. For strings starting with "248", the remaining 9 digits each have 10 possible values (0-9), so there are 9^10 such strings. For only even digit strings, each of the 12 digits has five possible values (0,2,4,6,8), so there are 12^5 such strings. Only even digit strings starting with "248" would have five possible values for the remaining 9 digits, so there are 9^5 such strings. There for the number of strings of 12 digits beginning with "248" or consisting of only even digits is 9^10+12^5-9^5, or 3,486,974,184 strings.