The formula based on Inclusion-Exclusion for the usual number $D(n)$ of derangements of $n$ objects can be generalized. The result is the following.

Fix $k\geq 1$. Let $\mathbb{N}=\lbrace 0,1,2,\dots\rbrace$. For $\alpha=(\alpha_1,\dots,\alpha_k)\in\mathbb{N}^k$, let $D(\alpha)$ be the number of fixed-point free permutations of the multiset with $\alpha_1$ 1's, $\alpha_2$ 2's, etc. Write $x^\alpha = x_1^{\alpha_1}\cdots x_k^{\alpha_k}$. Then
  $$ \sum_{\alpha\in\mathbb{N}^k} D(\alpha)x^\alpha =
       \frac{1}{(1+x_1)\cdots (1+x_k)\left(1-\frac{x_1}{1+x_1}-\cdots -
         \frac{x_k}{1+x_k}\right)} $$
  $$  = \frac{1}{1-\sum_S
        (|S|-1)\prod_{i\in S}x_i}, $$ 
where $S$ ranges over all nonempty subsets of $\lbrace 1,2,\dots,n\rbrace$. 

This result appears as Exercise 4.5.5 in Goulden and Jackson, *Combinatorial Enumeration*. It can be used to obtain a lot of information about $D(\alpha)$.