The equation can be inverstigated in the following way. First construct a row of 's: .
Now between any space you can place a break and that will give you a possible pair.
For example, to illustrate, , , we can break it as . That gives us . Thus, given as an integer we place breaks in between the breaks. There are a total of ways of doing this.
@Isomorphism: The proof of the combinatorial formula you used rests on what I did.
Hello, perash!
Here's a "visual" explanation of Isomorphism's solution . . .How many solutions does the equation: .
have on the set of positive integers?
We have a 100-inch "yardstick", marked in inches.
We will cut it into three pieces by making two cuts.
There are 99 inch-marks and we will chose two at which to cut.
Answer: . ways.