Let:
$k\in\Bbb Z$ ($\Bbb Z$ is the set of all integers)
$n\in\Bbb N$ ($\Bbb N$ is the set of all natural numbers (positive integers) and does not own/contain 0)
$\vec{c}\in\Bbb Z^n$ ($\Bbb Z^n$ is the cartesian product of the set $\Bbb Z$ with itself $n$ times)
$\vec p\in\Bbb R^n$ ($\Bbb R^n$ is the cartesian product of the set $\Bbb R$ with itself $n$ times)
$\Bbb R$ is the set of all real numbers.
All elements, terms, coordinates and components of vector $\vec p$ are greater than 0 and less than 1.
$\forall i\in\{1,\ldots,n\}:x_i\sim\text{Bernoulli}(p_i)$
How to compute and calculate: $\text{Probability}\Bigg(\displaystyle\sum_{i=1}^nc_i\cdot x_i=k\Bigg)$?
When all the elements of vector $\vec c$ are equal to 1, all the elements of vector $\vec p$ are equal to some real number in the unit interval $q$ and $k\in\{0,\ldots,n\}$ then the sum of all the $n$ bernoulli variables distributes binomial distribution with probability $q$ and number $n$ and then the probability in question is $\displaystyle\binom{n}{k}\cdot q^k\cdot(1-q)^{n-k}$, but I want to know the general and universal formula to compute and calculate the probability in question without the constraints and restrictions specified below the question.