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?
The equation solved by wolframalpha
(corrected equation in edit)
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 $n=68$...