What is the continuous distribution version of multinomial distribution?

A similar distribution would be the Dirichlet distribution.

A random sample of a Dirichlet distribution is a set of probabilities that add to one. You can then multiply each by, say, $24$, to get a "continuous multinomial distribution."
This is just a direct answer to your question about "continuous multinomial distribution", whether you should use it to model your data is another question.

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Statistics

Probability Distributions

Multinomial Distribution

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