I have a series of independent Binomial random variables, $X_1, X_2, \dots, X_n$. Here, $X_i$ has parameters $(n_i, p_i)$. I want to calculate the probability
$$P\left(\sum_{i=1}^n c_i X_i < c\right)$$
for some constants $c$ and $c_i$. I'll have to calculate this sum repeatedly with the $p_i$'s remaining the same but the $n_i$'s changing slightly every time. This involves getting all the $n$-tuples, $x_1, x_2, \dots , x_n$ that satisfy: $\sum c_i x_i < c$ and summing the PMFs across them.
This is an expensive affair since the number of such $n$-tuples will grow exponentially with $n$. Is there a way to calculate this efficiently that scales better with $n$ or approximate this probability in a way that it has better scaling characteristics?