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.
