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If I draw M repeated samples from Uniform Integer Distribution on [1,N], what will be the distribution of the number of times a given integer chosen among the integers 1,2,...,N appears in my M samples?

For concreteness, lets say that I draw repeatedly 100 times from Uniform Integer [1,5] distribution. What then will be the distribution of the number of times the integer 3 appears in my 100 samples?

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In a single draw, your chosen integer will be drawn with probability $\frac{1}{N}$. Since you draw with replacement, the total number of such successes is binomially distributed with parameters $M$ and $p=\frac{1}{N}$.

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