How many combinations of this tuple are possible?

Question

Given a tuple $(x,y,z)$ with $x,y,z \in \{1,...,10\}$ and $x<y-1<z-2$, how many combinations are possible? I know that $x$ can't be $10,9,8$ or $7$, but I am not sure how to calculate the exact number of possible combinations.

Answer 1

Take $10$ balls, label $3$ of them $x,y,z$, glue one ball each to the right of $x$ and of $y$ to enfore the inequalities, and then count the ways to choose $3$ spots for $x,y,z$ in the arrangements of the resulting $8$ objects.

Answer 2

Consider the range of values for $y=(x+1, 7)$ and the range of values for $z=(y+1, 10)$.

From these intervals, we deduce that for any $x\in\{1,...,7\}$, there are $7-x$ values for $y$. For any $y\in\{2,...,7-x\}$, there are $9-y$ values for $z$. The number of possible combinations is thus $$\prod_{x=1}^6\prod_{y=2}^{7-x}x(7-x)(9-y)$$ Plug this in Wolfram Alpha or do this by hand (if you prefer) should give you the answer.