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...how many beans will not be adjacent to the others, like this:

_o__o_o

...how many beans will be adjacent to one and only one bean, like this:

_oo_oo____oo_

...and what's the probability of having N "continuous" beans?

...and most importantly, why?

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Do you allow more than one bean per bin? In either case, you could think about how the stars and bars approach could help: en.wikipedia.org/wiki/Stars_and_bars_(combinatorics) –  Ross Millikan May 5 '12 at 4:26
    
By how many beans, do you mean the expected number (mean)? For the final question, do you mean exact expression (this can be obtained by inclusion/exclusion, but may not be useful), or an estimate? –  André Nicolas May 5 '12 at 4:51
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