I am reading the discussion in my probability book (graduate level Mathematical Statistics I course).
The framework is a lottery with 44 numbers. Here one needs to pick 6 numbers to win.
I understand that WITHOUT replacement and ordered the chance to win is 44!/38!
I get that WITH replacement and order, the chance to win is 44^6
I also get that, unordered and with replacement, the chance to win is 44!/(6!38!) because one is eliminating repeated groupings.
What I don't understand is UNORDERED WITH REPLACEMENT. The book provides the following description, but the concept of bins does not make sense to me. I really don't understand why they go from 44 slots to 49, and then divide by 43!. I've included a picture of the book section below. I was, like the book says, guessing that the odds of winning would be 44^6/6!
Could someone help explain this to me?
The framework is a lottery with 44 numbers. Here one needs to pick 6 numbers to win.
I understand that WITHOUT replacement and ordered the chance to win is 44!/38!
I get that WITH replacement and order, the chance to win is 44^6
I also get that, unordered and with replacement, the chance to win is 44!/(6!38!) because one is eliminating repeated groupings.
What I don't understand is UNORDERED WITH REPLACEMENT. The book provides the following description, but the concept of bins does not make sense to me. I really don't understand why they go from 44 slots to 49, and then divide by 43!. I've included a picture of the book section below. I was, like the book says, guessing that the odds of winning would be 44^6/6!
Could someone help explain this to me?