This problem is way over my head. I did the first 4 HWproblems in this section,

Question

This problem is way over my head. I did the first 4 HWproblems in this section, but this is one is like WHAT?
Anyway, any help would be appreciated. Will rate a lifesaver!
Problem: A calculator has an 8 digit display and a decimal point that islocated at the extreme right of the number displayed, at theextreme left, or between any pair of digits. The calculator canalso display a minus sign at the extreme left of the number. Howmany distinct numbers can the calculator display? (Note thatcertain numbers are equal, such as 1.9, 1.90, and 01.900, andshould therefore not be counted. This problem is way over my head. I did the first 4 HWproblems in this section, but this is one is like WHAT?
Anyway, any help would be appreciated. Will rate a lifesaver!
Problem:

Explanation / Answer

Now denote by |S_k| the number of elements in S_k. Since theintegers range from 0 to 99999999 on this calculator, we have S_0 =99999999 * 2 + 1 where the *2 is for negatives and the + 1 is tocount the number 0 which is the same whether positive ornegative.
To count S_k for k > 0, we note that the last digit cannotbe 0 as explained above. However, the 7 digits to the left of thelast digit can be anything you want them to be. (Try out anexample). Hence, for k > 0, |S_k| = 10^7 * 9 * 2 where the * 9is for the 9 possible last digits, and the * 2 is to countnegatives as well as positives.
Thus the number of numbers which the calculator can hold isSUM |S_k| = (99999999 * 2 + 1) + 8 * (10^7 * 9 * 2)= 1,639,999,999 assuming I didn't make any blunders.

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