Test a binomial series – MathOverflow

I tried to try

$ sum_ {r = 0} ^ {n} binom {n} {r} = sum_ {r = 0} ^ {n} frac {n!} {(r!) (nr)!} = 2 ^ n $

I used the introduction method:

Assuming $ sum_ {r = 0} ^ {n} frac {n!} {(r!) (n-r)!} = 2 ^ {n} $

Test $ sum_ {r = 0} ^ {n + 1} frac {{(n + 1)}!} {(r!) (n + 1-r)!} = 2 ^ {n + 1} $

I have $ sum_ {r = 0} ^ {n + 1} frac {{(n + 1)}!} {(r!) (n + 1-r)!} = sum_ {r = 0} ^ { n + 1} ( frac {n!} {(r!) (nr)!} + frac {n!} {(r-1)! (nr + 1)!}) $

And a strange term appeared: (-1)!

It's not valid

Can someone tell me how to prove this and what is wrong with the introduction form?