I want to prove the following or a similar result: For $1\le k \le n$ \begin{align}&1-\sum\limits_{j=k+1}^n\binom nj(1-x)^jx^{n-j}~~~~~~(1)\\ \underset{\text{wrong}}{\overset{\text{maybe}}=}~&1-\sum\limits_{j=n-k-1}^n\binom njx^j(1-x)^{n-j}~~(2)\end{align} I think that the equation as written here is wrong, but I want to get a result similar to this. (That means I want to transfer it to a sum with limits like in (2) or alternatively $\sum\limits_{j=n-k}^n$)
I tried applying the binomial theorem $$\sum\limits_{k=0}^n\binom nk x^{n-k}y^k=(x+y)^n$$ and the symmetry of the binomial coefficient $$\binom nj=\binom n{n-j},$$ but didn't yield any of the desired results, yet.