A number theory problem by Achal Jain

$\large x@y = x (x+1)(x+2) \cdots (x+y - 1)$

The operator $$@$$ is defined for positive integers $$x$$ and $$y$$ as above. Let $$h$$ denote the greatest common divisor of $$x @y$$ and $$y!$$. Which of the following options must be true?



Notation: $$!$$ is the factorial notation. For example, $$8! = 1\times2\times3\times\cdots\times8$$.

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