Finding the factorial of a lesser number from the factorial of a number.

Question

Given the factorial $F$ of a number $N$, how can one find the factorial of numbers lesser than $N$ with $F$, without computing the factorial of each lesser number by the formula $n! = n * (n - 1)!$

Answer 1

The factorial of $N$ is $N\cdot (N-1)\cdot (N-2)\cdots(N-k)\cdots 3\cdot 2\cdot 1$. So if $M$ is an integer less than $N$ that means $M=N-k$ for some integer $k$. Thus $M!= (N-k)\cdots 3\cdot 2\cdot 1 = \dfrac{N!}{N\cdot (N-1)\cdot (N-2)\cdots(N-k+1)}$.