Some algebraic identity and its relations to Stirling numbers of the second kind

In this short note we provide some algebraic identity with a proof exploiting its probabilistic interpretation. We show several consequences of the identity, in particular we obtain a new representation of a Stirling number of second kind, $$S(n,d)=\frac{1}{d!}\sum _{1\le j_1<j_2<\dots <j_{d-1}<n}1\cdot 2^{j_{d-1}-j_{d-2}}\cdots d^{j_1}$$ for integers $n\ge d$. Relating this to other known formula for $S(n,d)$ we also obtain $$\sum _{1\le j_1\le j_2\le \cdots \le j_{n-d}\le d}{j}_1{j}_2\dots ,j_{n-d}=d!\sum _{1\le j_1<j_2<\dots <j_{d-1}<n}1\cdot 2^{j_{d-1}-j_{d-2}}\cdots d^{j_1}\mathrm{.}$$ As a side effect, we have new proof of a known result stating that for any integer $d\in \mathbb{N}$ and any $x\in \mathbb{R}$ equality $$\sum _{r=0}^d(-1{)}^r\left(\genfrac{}{}{0px}{}{d}{r}\right)(x-r{)}^d=d!$$ holds. This is a special case of the presented identity.