Probability of the maximal marginal of a multinomial distribution

Question

So let us suppose I have a set of random variables described by the multinomial distribution:

$$ f(x_{1}, x_{2}, \ldots, x_{M}) = \frac{N!}{x_{1}! x_{2}! x_{M}!} \prod_{k=1}^{M} p_{k}^{x_{k}} $$

I want to find $P\left( \text{max} \left\{ x_{1}, x_{2}, \ldots, x_{M} \right\} < \gamma \right)$ for some value $\gamma \in \mathbb{R}_{++}$.

The problem is, that since we must have $\sum_{k=1}^{M} x_{k} = N$, the $x_{k}$ are not independent. Therefore, I can't use the simple order statistics rule:

$$ P\left( \text{max} \left\{ x_{1}, x_{2}, \ldots, x_{M} \right\} < \gamma \right) \neq \prod_{k=1}^{M} P(x_{k} < \gamma) $$

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Questions:

1. So what do I do instead to find the desired probability? Is it possible to even find this probability?
2. Could we instead upper and lower bound it?
3. Could we we relate it to $\prod_{k=1}^{M} P(x_{k} < \gamma)$ in terms of greater or less than?
4. Can we do any of these things if the distribution was homogenous, i.e. $p_{k} = p$ $\forall k$?