# Simplified Multinomial Distribution

I am working with a simple case of the multinomial distribution, as follows:

There are $k = 8$ different possible outcomes, each occurring with equal probability $p = \frac{1}{8}$.

What is the probability that, after $N$ trials, exactly $n$ of them resulted in outcome 1 (where $n$ might be any integer with $0 \leq n \leq N$)?

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The probability of outcome $1$ on each trial is $1/8$, so in this case you can lump the other outcomes into one category called "failure", and the question becomes: What is the probability of exactly $n$ successes in $N$ trials, with probability $1/8$ of success on each trial. The number of outcomes in just one category has a binomial distribution. So you get $$\binom N n \left(\frac 1 8\right)^n \left(\frac 7 8\right)^{N-n}.$$
Only if you'd specified the numbers of outcomes in all $8$ categories would you need the formula for the full-fledged multinomial distribution.