# Discrete Math Question

Prove the following theorem:
Theorem For a prime number p and integer i,
if 0 < i < p then p!/[(p− i)! * i] * 1/p

Not sure how to go about this. I wanted to do a direct proof and this is what Ive got so far.
let i = p-n
then p!/[(p-n)!*(p-n)] but that doesnt exactly prove much.

## Answers and Replies

RUber
Homework Helper
hlzombi,
First, please note that you should use the template and this seems like it might be better to place in the math forum.
Prove the following theorem:
Theorem For a prime number p and integer i,
if 0 < i < p then p!/[(p− i)! * i] * 1/p

Not sure how to go about this. I wanted to do a direct proof and this is what Ive got so far.
let i = p-n
then p!/[(p-n)!*(p-n)] but that doesnt exactly prove much.
##\frac{p!}{(p-n)!*(p-n)}=\frac{p*(p-1)*...*(p-n+1)}{p-n}##
Finally, I am not sure what you are asked to prove...is there some equality or property here?
You need to be more clear.

Apologies, I'm new here. I tried to follow the template as best as I could.

To clarify, I'm trying to prove the theorem p!/[(p− i)! * i] * 1/p where 0<i<p when p is a prime number and i is an integer.

pasmith
Homework Helper
Apologies, I'm new here. I tried to follow the template as best as I could.

To clarify, I'm trying to prove the theorem p!/[(p− i)! * i] * 1/p where 0<i<p when p is a prime number and i is an integer.

Your statement is incomplete. What are you trying to prove about $\frac{p!}{(p-i)!i} \times \frac 1p$ when $p$ is prime and $i$ is an integer?

Im trying to verify the theorem under those conditions

RUber
Homework Helper
You did not write the theorem.

RUber
Homework Helper
That says p divides p choose i. That is not what you wrote above.

RUber
Homework Helper
To demonstrate this, you can use induction.
Show that for a base case (i=1) ##p\left| \left( \begin{array}{c} p\\i\end{array}\right) \right. ##
Assume for some n < p-1, the statement holds.
Show that ##p\left| \left( \begin{array}{c} p\\n+1 \end{array}\right) \right. ##

Mark44
Mentor
Apologies, I'm new here. I tried to follow the template as best as I could.
When you use the template, don't delete its three parts.

Also, I moved this thread, as it was better suited in one of the math sections.