I am working on my discrete mathematics homework and I encountered this formula in my answer key. I wanted to know how to use it since I can't seem to find it in my textbook. The question for context is:
How many ways are there to have a collection of eight fruits from a large pile of identical oranges, apples, bananas, peaches, and pears if the collection should include exactly two different kinds of fruits?
Now in the answer key, to find the number of ways of selecting from the two fruit they use the following.
$x$ is the number of fruits of the first type, $y$ is the number of fruits of the second type
$$x + y = 8$$ $$x,y \ge 1 $$
and the following is what I do not understand
$${8-1-1+2-1}\choose 2-1$$
What exactly do the numbers in the following choose mean and where do they come from. I only as because if I encounter another problem of this sort I would like to know how to use the formula.
The answer is $${8-1}\choose 2-1 $$
For the first type of fruit you are free to choose whatever fruit you like. For the second one you can not choose the same fruit as the first one, so you have $8-1$ types and you are to choose $2-1$ type out of the remaining $8-1$ types.
The answer is $7$