Given $m$ balls in a container, having $n\le m$ colors, if balls are chosen from the container randomly without replacement until all balls have been removed and the order of choices is considered, what are the odds that a given "choice path" will contain no instances of two or more balls of the same color chosen consecutively?
By extension, given a number of colors $n$, what is the largest number of balls $m$ such that the odds of a random "choice path" containing no repeat choices is greater than $x$% (assuming that the colors are spread as evenly as possible, i.e., the count of balls for any particular color is no more than $1$ greater than the count of balls for any other color)?