Intuition behind division in permutation

Question

So, I am trying to explain this concept but I would like also to understand why it’s true.We have m objects and n spaces where $m\leq n$, what is the intuition behind the division between $n!$ and $(n-m)!$. I mean it works, considering two letters in four spaces we just need to multiply $4$ by $3$ and stop because there are no elements left, and algebraically this can be done also using the above formula, because common terms cancel out.But intuitively, why is that true?

Answer 1

To assign $m$ objects to $m$ spaces, we first assign $n$ objects to $m$ spaces ($x$ possibilities) and then assign the remaining $n-m$ objects to the remaining $n-m$ spaces. $$ m!=x\cdot (n-m)! $$ So what is $x$?

Answer 2

Suppose we have a standard 52 card deck of cards and we want to draw 3 cards. There are 52 * 51 * 49 permutations of this. This is naturally $\frac {52!}{48!}$. Ignoring suits for the moment, we draw a 10, J, 9. For each set of three cards, we have 3 * 2 * 1 or 3! possible arrangesments (ex. 10, 9, J; J 9 10; J 10 9 etc. etc.).

To neutralize the effects of each possible permutation, we would divide out the permutations. Therefore, we get $\frac {52!}{48!3!}$. Hope this helps!!