Getting rid of product of sequence sign

Question

I am having trouble with equation containing product of sequence:

$$\frac {1}{2} = 1 - \frac {\prod \limits_{i=1} ^{n} (366 - i)}{365^n} $$

How can I convert the $\prod \limits_{i=1} ^{n} (366 - i)$ part of the equation so that I can solve the equation?

(corrected equation in edit)

Are you sure of the equation ? It seems that no integer $n$ would satisfy it. — Claude Leibovici, Mar 07 '15 at 10:34
I assume you're working on the Birthday Problem, which shows that for $n=23$, the probability exceeds $\frac{1}{2}$. There's no integer solution. — GohP.iHan, Mar 07 '15 at 10:44
Yes, this is related to birthday problem indeed, but im not looking for the integer solution, the value that wolframalpha calculates ~ 22.7676903156182 is fine by me, how could I get to this on paper? — VOid, Mar 07 '15 at 10:50

score 1 · Accepted Answer · answered Mar 07 '15 at 11:03

So you can write

$$ \prod\limits_{i=0}^n (366 - i)= 366(366-1) \cdots (366 - n) = \frac{366!}{(366-n-1)!} $$

So your equation becomes

$$ \frac{365^n}{2} = \frac{366!}{(366-n-1)!} = \frac{366!}{\Gamma(366-n)} $$

Which you can solve perhaps by some numerical techniques.

By evaluating

$$ \frac{365^n\Gamma(366-n)}{2} = 366! $$

I found (one) solution is between $n=67$ and $n=68$...

1 Answers1