I do not know. By using binomial distribution formula, when I tried to calculate the probability of rolling the same number exactly $3$ times with $200 $ of seven hundred-sided dice, why the result of probability is more than one $(2.02)$ ? $$\binom{200}{3}\left(\frac{700}{700}\right)^1\left(\frac{1}{700}\right)^2\left(\frac{699}{700}\right)^{197}\approx 2.02$$
Help me please.
First it is not a valid binomial/multinomial formula so the value exceeds one. The corresponding probabilities have to be add up to $1$.
The probability that a specific number appear exactly $3$ times is $$\binom{200}{3}\left(\frac{1}{700}\right)^3\left(\frac{699}{700}\right)^{197}$$
If you want to calculate the probability that there is at least one number appearing exactly $3$ times, then you are calculating the union of $700$ of these events, which involve the use of multinomial and $66$ layers of inclusion-exclusion principle. Fortunately you can always truncate as what Bonferroni inequality did and also obtain a good precision.