The confusion regarding pigeonhole principle Below question and answer are from my textbook
Q: The largest $2$-digit number is $99 $. How many $2$-digit numbers must be in a set in order to apply the pigeonhole principle to conclude that there are two distinct subsets of the numbers whose elements sum to the same value? A: If there are n numbers, there are $2^n$ distinct subsets of the numbers. The sum of each subset is the range from $0$ to $99 n $, so the number of different possible values a sum can have is $ 99n + 1$ . In order to guarantee that two subsets have the same sum, the number of subsets must be larger than the number of different possible values for the sum.
The answer is the smallest number n such that
$$2^n > 99n + 1.$$
The inequality is satisfied for $n = 10$, but not $n = 9.$
I can't make sense of 'The sum of each subset is the range from $0$ to $99n$', suppose A is a set containing two 2-digit number, then the largest possible sum of a subset of A is
$$ 99 + 98 = 197$$
 and the smallest sum should be $10$(if the empty set is excluded), and all possible sums resulted from subsets of A should be integers all the way from $10$ through $197$ and $197$ is already the largest number we can get for a 2-element set containing only $2$-digit number, meaning there are 
$$197 - 10 + 1 = 188 \space \space \text{( excluding empty set for which the sum is 0)} $$
possible values as the sum of subsets of $A$. 
I get the gist of pigeonhole principle, just can't make sense of 'from $0$ to $99n$', assuming the upper limit is $99n$, in the case of set $A$, there will be two $99$ in a set $A = (99, 99)$, doesn't this contradict the definition of set that there shouldn't be any repetitive elements in a set?
My solution to this question is: To satisfy the requirement of applying the pigeonhole principle, the number of subsets of this set needs to be larger than the number of all possible sums resulted from all the subsets.
Then the idea can be expressed as(let k be the number of element(s) in the set):

$$2^k = \sum(99 + (99 - 1) + (99 - 2) + ... + (99 - k + 1)) - 10 + 1 \space \space \text { (excluding empty set here)}$$
  Although the answer will also be $10$ as shown in the reference, I just feel the solution my textbook offers is somewhat flawed, or am I not understanding the answer correctly somewhere?

Any assistance would be appreciated! 
