Penalty kicks with a shifting rate

You take $100$ penalty kicks. You make the first and miss the second. From then on, the probability you score any kick equals the fraction of kicks you've made so far (e.g. after $17$ makes in $30$ attempts, the next has probability $\tfrac{17}{30}$). The probability that you finish with exactly $66$ made can be written as $\tfrac1M$ for an integer $M$; find $M$.