Any given day will be either clean or clouded, but not both. This means that for one single day, we have two choices- 1. Clean, 2. Clouded.

So number of choices for one day= 2 (or 2^1)

Number of choices for n days= 2^n (2 choices for every day)

Number of choices for 7 days= 1 week = 2^7 which is equal to the number of ways