Summation of some factorial terms

Question

I have an interesting series whose sum I know will be $0$, but not able to prove it. The series is

$$\sum_{i=0}^r \frac{(-1)^{i}}{i! (r-i)!}=0.$$ I had checked it for several values of $\text {“}r\text{''}$ and got the result, but was not able to show it. I got this observation while proving the Leibniz formula in case of distributions.

Any type of help will be appreciated. Thanks in advance.

Answer 1

Because its just $r!(1-1)^r=0$

Answer 2

Multiply your sum by $r!$ You will obtain :

$$\sum_{i=0}^r \binom{r}{i} (-1)^{i}(1)^{n-i}$$

which is $0$ because it is the binomial expansion of $(1-1)^r.$