Let's study the n-tuple $\displaystyle X=(X_1,\dots,X_n)$

This follows the same distribution as $\displaystyle X'=(7-X_1,\dots,7-X_n)$

This means that for any measurable application h, we have $\displaystyle \mathbb{E}(h(X))=\mathbb{E}(h(X')) \quad {\color{red}\star}$

Let :

$\displaystyle h\equiv h_a~:~ \mathbb{R}^n\rightarrow \mathbb{R}$

$\displaystyle h_a((x_1,\dots,x_n))=\begin{cases} 1 \quad \text{if }x_1+\dots+x_n=a \\ 0 \quad \text{otherwise} \end{cases}$

It is obvious that :

$\displaystyle \mathbb{E}(h_a(X))=\mathbb{P}(X_1+\dots+X_n=a)$

and

$\displaystyle \mathbb{E}(h_a(X'))=\mathbb{P}(7-X_1+\dots+7-X_n=a)=\mathbb{P}(X_1+\dots+X_n=7n-a)$

By $\displaystyle \color{red} \star$, we get $\displaystyle \boxed{\mathbb{P}(X_1+\dots+X_n=a)=\mathbb{P}(X_1+ \dots+X_n=7n-a)}$

Which is a generalization of your situation (n=1000, a=1100)