TITLE: Difficultly understanding the inclusion exclusion identity because of difficult notation
QUESTION [0 upvotes]: This is the property: $\mathbb{P}(E1\cup E2\cup ...\cup En)= \sum_{i}^{n} \mathbb{P}(E1) - \sum_{i1<i2} \mathbb{P}(Ei1Ei2) + ... + -1^{r+1}\sum_{i1<i2<...<ir} \mathbb{P}(Ei1Ei2...Eir) + -1^{r+1}\sum_{1<2<...<n} \mathbb{P}(E1E2...En) $
While I understand the property, I don’t understand the summation symbol notation. What does the book mean by $\sum_{i1<i2<...<ir}$? Can someone draw out specifically what this sum is so I can understand it? I lost my question privileges due to this question but can someone please let me know what about this question was so bad? I couldn't find the syntax anywhere.

REPLY [0 votes]: Some people would write the sum $\sum_{i_1<i_2<\cdots<i_r}$ as
$\sum_{1 \leq i_1<i_2<\cdots<i_r \leq n}$
in order to emphasize that all the variables $i_1,i_2,\ldots,i_r$ have to be between $1$ and $n$ inclusive.
The $r$th sum can be rewritten in more basic summation notation as
$$
\sum_{1 \leq i_1<i_2<\cdots<i_r \leq n} \mathbb{P}(E_{i_1}E_{i_2}\cdots E_{i_r})
= \sum_{i_1 = 1}^{n - r + 1}\, \sum_{i_2 = i_1 + 1}^{n - r + 2}\,
   \sum_{i_3 = i_2 + 1}^{n - r + 3} \cdots  \sum_{i_{r-1} = i_{r-2} + 1}^{n - 1}\,
   \sum_{i_r = i_{r-1} + 1}^{n} \mathbb{P}(E_{i_1}E_{i_2}\cdots E_{i_r}).
$$
Each side of the equation ends up with $\binom nr$ individual terms in the sum, one for each possible sequence of $r$ increasing integers where none is less than $1$ or greater than $n.$
Each of those sequences picks out a different subset of $r$ of the events in the set $\{E_1, E_2, \ldots, E_n\}.$
Personally, I find the notation on the left is a lot easier to understand.
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