How many solutions does the equation
$\displaystyle x+y+z=100$
have on the set of positive integers?
The equation $\displaystyle x_1+x_2+...+x_k = n$ can be inverstigated in the following way. First construct a row of $\displaystyle n$ $\displaystyle 1$'s: $\displaystyle 1 \ 1 \ 1 ... \ 1$.
Now between any space you can place a break $\displaystyle \bold{b}$ and that will give you a possible $\displaystyle (x_1,...,x_k)$ pair.
For example, to illustrate, $\displaystyle n=5$, $\displaystyle k=3$, we can break it as $\displaystyle 1 \ 1 \ \bold{b} \ 1 \ \bold{b} \ 1 \ 1$. That gives us $\displaystyle x_1=2, x_2=1,x_3=2$. Thus, given $\displaystyle n$ as an integer we place $\displaystyle k-1$ breaks in between the $\displaystyle n-1$ breaks. There are a total of $\displaystyle {{n-1}\choose{k-1}}$ 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: .$\displaystyle x+y+z=100$
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: .$\displaystyle {99\choose2} \:=\:4851$ ways.