[R] expand.grid game

[baptiste auguie]

It did take me a good night's sleep to understand it. I was stuck with
the exact same question but I see now how the remaining balls are
shared among all 8 urns (therefore cases with 11, 12, 13, ... 17 balls
are also dealt with).

Thanks again,

baptiste


2010/1/12 Rolf Turner <r.turner at auckland.ac.nz>:
>
> 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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