Decrease order of multi-index in a sum..

Decrease order of multi-index in a sum..

can anyone help me writing the sum, $$\displaystyle
\sum_{|\beta|=M+1}\frac{1}{\beta!},$$ as a sum whose multi-index have
order $M$.
Ilustration: Since $\beta$ has length $M+1$ one can find a multi-index
$\alpha$ with length $M$ such that $\beta=\alpha+\delta_j$ for some $j\in
\{1, \ldots, n\}$. Someday I thought I could write the above sum as,
$$\sum_{|\alpha|=M}\sum_{j=1}^n \frac{1}{(\alpha+\delta_j)!},$$ but I
realized this last sum has more terms the the first one, I'll illustrate
below the reason for this:
For example in $\mathbb N_0^3$ where $\mathbb N_0=\mathbb N\cup\{0\}$ the
multi-indices of length 3 are: $$(3, 0, 0), (0, 3, 0), (0, 0, 3), (2, 0,
1), (2, 1, 0), (0, 2, 1), (0, 1, 2), (1, 0, 2), (0, 1, 2), (1, 1, 1),$$
whereas the multi-indices of length $2$ are $$(2, 0, 0), (0, 2, 0), (0, 0,
2), (1, 0, 1), (1, 1, 0), (0, 1, 1).$$ Notice one can get the
multi-indices of length 3 from those of length 2 summing $\delta_1,
\delta_2, \delta_3$, but we don't need all multi-indices of length 2 to do
this..
I don't know how to solve this problem... Any help will be valuable...Thanks