There were four portions of warden pie and four portions of venison pasty
to be distributed among eight out of eleven guests. But five out of the
eleven will only eat the pie, four will only eat the pasty, and two are
willing to eat of either. Any possible combination must fall into one of
the following groups. (i.) Where the warden pie is distributed entirely
among the five first mentioned; (ii.) where only one of the accommodating
pair is given pie; (iii.) where the other of the pair is given pie; (iv.)
where both of the pair are given pie. The numbers of combinations are:
(i.) = 75, (ii.) = 50, (iii.) = 10, (iv.) = 10--making in all 145 ways of
selecting the eight participants. A great many people will give the
answer as 185, by overlooking the fact that in forty cases in class
(iii.) precisely the same eight guests would be sharing the meal as in
class (ii.), though the accommodating pair would be eating differently of
the two dishes. This is the point that upset the calculations of the
company.