number of ways to make a selection

Question

In a book I saw a paragraph , but I cannot understand what is the logic behind that

Can anybody explain me this .

Answer 1

Lets start with a slightly different question: How many ways are there to take none or more items? If there is only one kind of item, and p of that item, obviously you can take 0, 1, 2, ..., p of them, so there are (p+1) ways of taking none or more. If there are two kinds of items, p of the first and q of the second, then whatever you take will have none-or-more of the first and none-or-more of the second, so you can reinterpret this as taking none-or-more of the first, THEN none-or-more of the second, independently of each other, and that independence implies a product, (p+1)(q+1). The generalization to more kinds is obvious. Now your original question is exactly the same except that you are not allowed to NOT take any items, so you have to subtract 1 to NOT count the case where you take none of any of them.

Answer 2

The 1 in the formula comes from the choice of 0 things of the respective kind.