I was going through the text An Introduction to Formal Language and Automata by Peter Linz where I came across an excerpt:
$$|y| = k \leq m.$$ We then look at $xz$ which has length $m! — k$. This string is in L only if there exists a $j$ such that $$m! — k = j!$$ But this is impossible, since for $m > 2$ and $k \leq m$ we have $$m! - k > (m - 1)!$$
But I do not quite understand their logic of the equation $m!-k=j!$ being impossible.
What came to my mind is something like this:
$$m!-k= \{m.(m-1).(m-2)...k....4.3.2.1\} -k $$ $$= k.(\{m.(m-1).(m-2)...(k+1)(k-1)....4.3.2.1\} -1) $$
which cannot be written in any factorial notation. But this way is very much intuitive and I do not quite like it.
Please can any one help me with the approach said in the text.
I think that the explanation beneath is quite clear? The distance between $m!$ and the greatest possible value for $j!$ (which is $(m-1)!$) is quite large you see. $m!-j! \geq m!-(m-1)! = (m-1)(m-1)!$ which is trivially greater than $m$ for $m>2$ as the explanation said. So $m!-k$ can't quite reach other factorial values underneath.