Counting Number of Objects - When to Add One Back I know this might be a very basic question. Sometimes to count objects, we just subtract. For example -- If there are 5 apples and I take away 1, then remaining are $5 - 1 = 4$ apples.
But other times, we have to add one back.
For example -- the number of days from the 1st to the 5th of any month = $5 - 1 + 1 = 5$. I added the last one back because I'm including the 1st.
What are these properties formally called? Are there more cases? Are there more formal explanations and properties of these cases in combinatorics?
 A: It's a little bit subtle.
When we think about apples we say that $5$ apples is four more apples than $1$ apple.  Because we're comparing the set of 5 whole apples to one whole apple.
When we think about days you're comparing the end of the first of June to the beginning of the fifth of June.
So the difference in days is only three.
So in fact it's a question of language not mathematics.  If your train is leaving Ambridge, and it stops at Brumley, Catford, and Dronby before arriving at Endton where you get off. You might say there are four stops until Endton (Brumley, Catford, Dronby and Endton) or you might say there are three (Brumley, Catford and Dronby).  It depends which country you come from.  In England we say four, in many European countries they say three. I haven't done an exhaustive survey. Most people are pretty convinced their way is right, but it's just convention.
So the only difference is whether the last item is included. Which differs by convention depending what the items are.  As mathematicians we think imprecision is bad so we don't talk about this sort of thing.
A: The number of days from the 1st to the 5th is only $5$ depending on how you define "from." How many days is it "from" 1st to the 2nd? Most people would say $1$ day, not $2$ days.
This is not a case of subtraction, but rather a case of counting a range - you are looking for the number of days between the 1st and 5th, inclusive.
I suppose one could say this is a difference between using cardinal numbers and ordinal numbers, but I think the real problem is your use of the word "from" as a synonym for subtraction, always. It is not, when it is dealing with a range.
