I was playing around with hypergeometric probabilities when I wound myself calculating the binomial coefficient $\binom{10}{3}$. I used the definition, and calculating in my head, I simplified to this expression before actually calculating anything
$$
\frac {8\cdot9\cdot10}{2\cdot3} = 120
$$
And then it hit me that $8\cdot9\cdot10 = 6!$ and I started thinking about something I feel like calling generalized factorials, which is just the product of a number of successive naturals, like this
$$
a!b = \prod_{n=b}^an = \frac{a!}{(b-1)!},\quad a, b \in \mathbb{Z}^+, \quad a\ge b
$$
so that $a! = a!1$ (the notation was invented just now, and inspired by the $nCr$-notation for binomial coefficients). Now, apart from the trivial examples $(n!)!(n!) = n!$ and $a!1 = a!2 = a!$, when is the generalized factorial a factorial number? When is it the product of two (non-trivial) factorial numbers? As seen above, $10!8$ is both.
Number Theory – Factorial Equations A! B! = C! and A! B! C! = D!
elementary-number-theoryfactorial
Related Solutions
If the factorials in the limit expression differ by an integer, then cancel away.
But if the difference only becomes an integer in the limit, then you obviously cannot cancel before taking the limit, so you are not allowed to cancel at all.
The difference between $x$ and $2x$ is $x$ which is not an integer until the limit of $-1$ is reached.
One way is the total number of leaves of a (single) rooted tree in which each leaf is minimally linked to the root by exactly $n-1$ edges, and which has the following property: the root has $2$ children, each child of the root has $3$ children, each child of each child of the root has $4$ children, and so on until the leaves are reached. A natural term for this is factorial tree, but I don't know if this phrase is in general use for this notion.
For example, for $n = 4$:
Best Answer
See Chris Caldwell, The diophantine equation $A!B!=C!$, J. Recreational Math. 26 (1994) 128-133. $9!=7!3!3!2!$, $10!=7!6!=7!5!3!$, and $16!=14!5!2!$ were the only known non-trivial examples of a factorial as a product of factorials as of the 3rd edition of Guy, Unsolved Problems In Number Theory (Problem B23).