Try to finish writing down the solutions of your quadratic equation, and check if it makes sense to put them back in your initial problem with factorials...
I expanded this to:Prove that n has no solutions given that r=2 when:
Which cancels down to:
Which goes to:
Which goes to:
This is now a quadratic which, when treated with the discriminant (a=1, b=-1, c=-8)
So, this quadratic has real solutions, but the questions asks to prove it does not. What have I done wrong?
Thanks.
is the product of the first positive integers. So the usual definition makes no sense if is not an integer and your teacher is fully right.
How would one anyway generalize it for any real number in a "nice way"? Two answers:
- in fact, you are considering a binomial coefficient and these still make sense if is any real number, since: (the number of terms is , which is an integer).
- refering to my last post, you can use Euler's Gamma function to extend the factorial, like Google does. This is a function defined on by (well, it can even by defined for other real or even complex numbers, but let's keep simple). Using integration by part, one can show that for every , . So this function is a candidate for a generalization of the factorial. In fact, for every , , and is a smooth, convex (even log-convex) function. As a conclusion, under various requirements, appears to be the most natural extension of the factorial: . But usually we write it with and not with the exclamation mark "!".