canceling factorials.

and what I mean by canceling later is my teacher said to do this
I dont know why you cant cancel the factorial function at the point where you have one on each side where I noted. I do not see the why the answer should be different when you cancel it later. Could someone explain the rule for canceling !'s thanks.
example

10Pn=90
10!/(10-n)!=90
10!=90(10-n)! *****I want to know why you cant cancel here.
10!/90=(10-n)!
8!=(10-n)!****but you can cancel here
8=10-n
n=2

but if you canceled at 10!=90(10-n)! you have n= -.888888

\(\displaystyle \begin{gathered} \frac{{10!}}{{\left( {10 - n} \right)!}} = 90\, \hfill \\ \,\frac{{10!}}{{90}} = \left( {10 - n} \right)! \hfill \\ \frac{{10!}}{{10 \cdot 9}} = \left( {10 - n} \right)! \hfill \\ 8! = \left( {10 - n} \right)! \hfill \\ \, \Rightarrow \,n = 2 \hfill \\ \end{gathered}\)

I know but why cant you cancel the ! earlier? I understand you can at the last step but I dont see why you cant earlier, why does the answer come out different?

brentwoodbc said:

I know but why cant you cancel the ! earlier? I understand you can at the last step but I dont see why you cant earlier, why does the answer come out different?

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One can never cancel the factorial sign (!).
WHY? Well it is not a number, nor does it stand for a number.
In the last step \(\displaystyle 8!=(10-n)!\), I simply realized the \(\displaystyle n=2\) is the only value that makes the statement true. There was no ‘canceling’ any thing there.

It is true that \(\displaystyle \frac{n!}{(n-3)!}= \frac{(n)(n-1)(n-2)(n-3)!}{(n-3)!}= (n)(n-1)(n-2)\).
But still there was no ‘canceling’ of the factorial sign.
We divided out the factor \(\displaystyle (n-3)!\)

ok ya that makes sense, my teacher said "you can cancel here now" which confused me

"Factorial" in this sense, is a 1-1 function on the natural numbers (not including 0).

If F(n) = n!, we can do the following F(n)=F(m) => n=m. That is the meaning of a one-to-one function. So in a sense you can "cancel" a factorial only when you are given a situation like the above. You cannot cancel them in the familiar way with division.

brentwoodbc said:

10!=90(10-n)! *****I want to know why you cant cancel here.

Click to expand...

What would you be cancelling with what?

Thank you!