# binomial conditional distribution. A question about a process

As you can see from the image below, how does Px,z(x,z) change to Px,y(z-x)? Can someone tell me the logic behind it or, refer me to a link that talks about the similar topic? Thanks alot!!

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Note that $Z=X+Y$. So we have $(X,Z)=(x,z)$ if and only if $(X,X+Y)=(x,z)$. But $X=x$ and $X+Y=z$ iff $X=x$ and $Y=z-x$.
Suppose that $X=5$ and $X+Y=17$. What is $Y$? (Of course I should say suppose $X$ has taken the value $5$, and $X+Y$ has taken the value $17$. But I wanted to simplify the statement.) –  André Nicolas Dec 1 '12 at 3:38