How would one simplify the product
$\prod_{i=1}^{N} (a_{i} + 1) $ ? I know that three of the terms would be $ \prod_{i=1}^{N} a_{i} $, $\sum_{i=1}^{N} a_{i} $ , and $1$ . The other terms seem to result from permutations, so would this involve cycles? Thank you for your time.