On 13/01/2010, at 9:19 AM, Greg Snow wrote:
How trivial is probably subjective, I don't think it is much above
trivial. ?I would not have been surprised to see this question on an exam in
my undergraduate (300 or junior level) probability course (the hard part was
remembering the details from that class from over 20 years ago). ?My
favorite test question of all time came from that course: "You have a deck
of poker cards with the 3's removed (and jokers), you deal yourself 5 cards
at random, what is the probability of getting a straight (not including
straight flushes)?"
This problem is simpler. ?Just think of the 8 places in the number as
urns, and the 17 1's as balls to be put into the urns. ?One ball has to go
in the first urn, so you have 16 left, there are choose(16+8-1,8-1) ways to
distribute 16 undistinguishable balls among 8 distinguishable urns. But that
includes some solutions with more than 9 balls in an urn which violates the
digits restriction, so subtract off the illegal counts. ?If we place 10
balls in the first urn, then we have 7 remaining balls to distribute between
the 8 urns or choose( 7+8-1, 7), If we place 1 ball in the first urn and 10
balls in one of the 7 other urns (7*), then there are choose( 6+8-1, 7 )
ways to distribute the remaining 6 balls in the 8 urns. ?Not too complicated
once you remember (or look up) the formula for urns and balls.
Sorry to be a thicko --- but doesn't the foregoing solution *leave in* the
possibility
of putting all 17 balls in the first urn? ?Or 3 balls in the first urn, 12
in the second,
and the remaining 2 in any of the other six urns? ?Etc. ?I.e. don't more
terms have to
be subtracted?
? ? ? ?cheers,
? ? ? ? ? ? ? ?Rolf Turner
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