"In base \(b\), there are exactly one hundred three-digit numbers whose digits are all distinct. (That's "one hundred" in the ordinary sense, \(100_{10}\).)

What is \(b\)?"

I'm not sure how to solve this. Thanks!

Guest Jun 15, 2023

#1**0 **

There are b3 three-digit numbers in base b. If all of the digits are distinct, then the first digit can be chosen in b−1 ways, the second digit can be chosen in b−2 ways, and the third digit can be chosen in b−3 ways. So, there are (b−1)(b−2)(b−3) three-digit numbers in base b with distinct digits. We are given that (b−1)(b−2)(b−3)=100. Solving for b, we find **b=13.**

Guest Jun 15, 2023