Is There a Way to Specify Limits On a Factorial

Question

If I want to be able to express a factorial -- let's say "$20$!" -- but with upper and lower limits such that the factorial is evaluated from Upper Limit, $n_1=20$, through a Lower Limit, $n_2=10$, for example, is there a convenient mathematical function that performs that?

Obviously, a simple subtraction operator won't do much since twenty-factorial from $n_1=20$ to $n_2=10$ does not equal either of the two possible subtraction quantities you could use:

*** $20!$ [Conditions: Evaluated from $n_1=20$ through $n_2=10$]

$= (20 \times 19\times 18 \times \cdots \times 10)$ does not equal $(20-10)!$ or $(20!)-(10!)$

I'm very interested in the use of such a function. Obviously, a used lower valued numbers for simplicity but, in practice, I'm not always afforded such nice numbers and I need an easy way to express a restricted factorial.

Answer 1

A non standard notation is $nPr$ where $nPr$= $n!/r!$, which is what you want.

Answer 2

You know from the definition of factorial, that $n! = n\cdot (n-1)!$ . Therefore, you have

\begin{array} 2\frac{5!}{2!} &= \frac{5\cdot 4\cdot 3 \cdot 2!}{2!}\\ &=5\cdot 4\cdot 3 \end{array} as an example.

In general, if $r<n-1$

$$\frac{n!}{r!} = n \cdot (n-1) \cdot ... \cdot (n-r+1)$$

Answer 3

There are a number of different notations for the rising and falling factorials; none appears to be completely standard. Frequently one sees the falling factorial written using the Pochhammer symbol: $$ (x)_n = x(x-1)(x-2)\cdots(x-n+1). $$ Here the subscript $n$ denotes the number of terms in the product, so your example is $(20)_{11}$. Of course, the rising factorial, $$ x^{(n)}=x(x+1)(x+2)\cdots(x+n-1), $$ can express the same product: $10^{(11)}=(20)_{11}=20!/9!$.