Correct Solution to Problem 1: While it is possible to do subtraction to correct for the over-counting that has occurred above, it is easier to simply take care to avoid over-counting in the first place. One might observe, for example, that the trouble described in the analysis above does not arise on flags whose last two stripes are colored red or green. The number of such flags (flags with exactly six blue stripes) is precisely C(8,6)*2*2 = 28*4 = 112. Next, the number of flags that have exactly seven blue stripes can be counted. There are C(8,7) ways to choose seven stripes to color blue, and the last stripe must be either red or green. Hence, there are C(8,7)*2 = 8*2 = 16 such flags. Finally, there is one flag with all eight stripes blue. Using the Sum Rule, the total number of flags with at least six blue stripes is 112 + 16 + 1 = 129.