וחלקום והעמידום על עשרים וארבעה בללום ונתנום בקלפי בא ידעיה ונטל חלקו וחלק חבריו שש בא [חרים] ונטל חלקו וחלק חבריו שש וכן פשחור וכן אימר

How fitting that this גמרא falls out in פרשת פינחס. In an installment of על פי חשבון, I attempted to calculate the probability of the Divinely "rigged" גורל falling out exactly like it did. So, then, what would be the odds of this one coming out like it did without the obvious Divine intervention?

I have to admit, I'm not sure I fully understand the exact process and exactly who picked what. I also have not investigated the מפרשים sufficiently. So I will keep it as simple as possible. Therefore, we will assume that the one picked for each משמר did not have to pick the name of the משמר first. In other words, ידעיה picked first. He came up with 6 names, all of which belonged to the ידעיה family but one was actually named ידעיה. Let us assume that he picked all 6 at the same time*. Using simple

*nCr*notation, there would be

*different possible combinations for ידעיה. For the uninitiated, that is simply a fanyc way of expressing the following: 24! / (24-6)! x 6! Simply put, the different possible combinations ar*

_{24}C_{6}24 x 23 x 22 x 21 x 20 x 19 |

6 x 5 x 4 x 3 x 2 x 1 |

I probably have not sufficiently explained the above concept for those who are not familiar but if we move on, I believe the probability can be expressed thusly:

1 |

x _{24}C_{6} x _{18}C_{6}_{12}C_{6} |

1 |

134596 x 18564 x 924 |

*Well I just looked a little deeper into this.:רש"י ערכין י"ב explains that the raffle happened exactly the way I just assumed that it didn't. So first the head of the משמר picked his own name and then the other 5. This complicated things slightly. The probability would therefore be:

1 |

24 x x 18 x _{23}C_{5} x 12 x _{17}C_{5}_{11}C_{5}_{} |

You want to simplify all this? According to רש"ש, there was nothing particularly miraculous going on here at all. There were in fact 4 separate raffles of 6 whereby each of the 4 משמרות would draw to determine the order of their respective subdivisions.

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