Most likely counts of outcomes of a multinomial after N trials (mode of multinomial) Given a categorical random variable $X = \{x_1,x_2,...,x_k\}$ with corresponding probabilities of each outcome $P = \{p_1, p_2,...,p_k\}$ (i.e. a multinomial):

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*What are the most likely counts of outcomes aftern $N$ trials?
Let's call the vector of counts after $N$ trials a state: $C_N = \{c_1^N, c_2^N,...,c_k^N\}$. What is the most likely observable state after $N$ trials? In an observable state counts must be integers.
If you just multiply $x_i \times p_i$ you get fractional counts, that doesn't work. Is there a better strategy than computing $x_i \times p_i$ values and then trying all roundings where $\sum_i c_i^N = N$?

*What is the next most likely state (the one that's a little less likely than the best one)?
An example:
Outcome probabilities: $[p_1=0.60, p_2=0.30, p_3=0.10]$.
Number of trials: $N=7$.
Most likely state for $N=7$ trials: $[c_1^7=4.20, c_2^7=2.10, c_3^7=0.70]$.
If you just round it to: $C_{round}=[4, 2, 1]$, you get $P(C_{round})=P([4, 2, 1])=0.1225$, where $P(x)$ is a probability function.
However, $P([5, 2, 0])=0.1470$. So just rounding doesn't work. I guess because of the multinomial coefficient getting in the way of computing probabilities.

You could start with a solution rounded down. For instance in the example you start with [4,2,0]. Then you iteratively keep adding the remainder (from rounding down) in the places that are making the best improvement.
This is my intuition. I still have to prove that this works.
