Solutions for Basic Counting and Listing

CL-1.2 We can form n digit numbers by choosing the leftmost digit AND choosing the next digit AND · · · AND choosing the rightmost digit. The first choice can be made in 9 ways since a leading zero is not allowed. The remaining n − 1 choices can each be made in 10 ways. By the Rule of Product we have 9× 10n−1. To count numbers with at most n digits, we could sum up 9× 10k−1 for 1 ≤ k ≤ n. The sum can be evaluated since it is a geometric series. This does not include the number 0. Whether we add 1 to include it depends on our interpretation of the problem’s requirement that there be no leading zeroes. There is an easier way. We can pad out a number with less than n digits by adding leading zeroes. The original number can be recovered from any such n digit number by stripping off the leading zeroes. Thus we see by the Rule of Product that there are 10 numbers with at most n digits. If we wish to rule out 0 (which pads out to a string of n zeroes), we must subtract 1.