here are P=25100 possible strings of 25 characters. Let's write this as P=5200. We need to know how many digits are in this number. We can try to solve the equation 5200=10n to give us an idea of how many digits are in P, because 10n−1≤P<10n would mean P has exactly n digits. How do we make use of the clue? Let 5200=(102)200=102002200, so we want to solve 102002200=10n which becomes 2−200=10n−200 Now we take log10 both sides, and get −200log10(2)=n−200 and, using the approximation log10(2)≈0.3 gives us −60=n−200⇒n=140 Our answer is 140.