Multinomial: Probability of a Subset I have been trying to calculate the probability of getting only a certain subset of  events of a multinomial distribution given that a certain number of events occur.  
That is, I have a multiniomial distribution 
$$ 
f(x_1, x_2, ... ,x_k; n, p_1, p_2, ... ,p_k) = Pr[X_1 = x_1, X_2 = x_2, ..., X_k = x_k] =  \frac{n!}{x_1! x_2!...x_k!} {p_1}^{x_1}  {p_2}^{x_2} ... {p_k}^{x_k}
$$
I want to calculate, as a example, 
$$ Pr[X_1 >0,   \mbox{ } X_2 >0,  X_5 > 0,   \mbox{ }  X_3 = 0,  \mbox{ } X_4 = 0,  \mbox{ } X_6 = 0,  \mbox{ } X_7 = 0, ..., X_k = 0  \mbox{ }| n = 8] $$
More generally, I want to be able to calculate the probability that event occurs such that $\{X_i, X_j, X_k\} >0$ and $\{X_m =0 \}$ for $m \not\in \{i,j,k \}$ given $n$.  
Is there a way to do this? I only really need to be able to calculate it numerically, but I'm finding even that surprisingly difficult to accomplish. 
 A: For a chosen $S\subseteq\{1,2,\dots,k\}$ you want a way to calculate:
$$P(X_i>0\text{ if and only if }i\in S\})$$ 
or equivalently:$$P(S=\{i\mid X_i>0\})$$
First let me focus on the example you mention where $n=8$ and $S=\{1,2,5\}$.
Then with inclusion/exclusion we find that the probability equals:$$P(X_1+X_2+X_5=8)-P(X_1+X_2=8)-P(X_1+X_5=8)-P(X_2+X_5=8)$$$$+P(X_1=8)+P(X_2=8)+P(X_5=8)=$$$$(p_1+p_2+p_5)^8-(p_1+p_2)^8-(p_1+p_5)^8-(p_2+p_5)^8+p_1^8+p_2^8+p_5^8$$

Addendum:
An expression for the general case is:$$P(\hat S=S)=\sum_{T\in\wp (S)}\left (\sum_{i\in T}p_i\right)^n (-1)^{|S|-|T|}$$where $\hat S$ denotes random set $\{i\mid X_i>0\}$ and $S$ denotes a fixed subset of $\{1,\dots,k\}$.
Finding this starts with the equality:$$P(\hat S=S)=P(\hat S\subseteq S)-P\left(\bigcup_{i\in S}\{\hat S\subseteq S-\{i\}\}\right)$$
where $P(\hat S\subseteq S)=P\left(\sum_{i\in S}X_i=n\right)=\left(\sum_{i\in S}p_i\right)^n$ and for the other term we find an expression by applying the principle of inclusion/exclusion.
