Consider $A_{1} \dots A_{n}$ some event and let's denote $\displaystyle S_{i} = \sum_{J_{r}}\mathbb{P}(A_{k_{1}} \cap \dots \cap A_{k_{r}})$, where $J_{r} = \{[k_{1},\dots , k_{r}] | k_{i} \ne k_{j},i \ne j\}$ (non-ordered subsets).
Let $B_{m}$ be event , which told us that occur only $m$ events from $A_{1}\dots A_{n}$.
How to prove that $\displaystyle \mathbb{P}(B_{m}) = \sum_{r = m}^{n}(-1)^{r-m}\binom{r}{m}S_{r}$ ?
Actually I thought about inclusion-exclusion principle. I guessed :
$\displaystyle \mathbb{P}(B_{m}) = \sum_{J_{m}}\mathbb{P}(A_{j_{1}} \cup \dots \cup A_{j_{m}}) = \sum_{J_{r}}\left (\mathbb{P}(A_{j_{1}}) + \dots+\mathbb{P}(A_{j_{m}}) - \dots \right)$. After I thought about represent sum of all $\mathbb{P}(A_{q})$ like $S_{1}$ , and there will be $\binom{n-1}{m-1}$ as a coefficient before $S_{1}$. And for all $S_{i}$ coefficient will be $\binom{n-i}{m-i}$ and that's true for all $1 \le i \le m$. So , I have that : $\displaystyle \mathbb{P}(B_{m}) = \sum_{k = 1}^{m}(-1)^{k-1}S_{k}\binom{n-k}{m-k}$, but it's not true.
Where is my fault? Any idea to proof?