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The maximum number of combinations is all seven. Two black-white-and-tan puppies and one of each other colour combination satisfies the conditions.
The minimum number of combinations is three.
One can not be the answer, because then, eight are of one or more colours which contradicts the premise that there are five of each colour.
Two can not be the answer, because there is just no way to split up eight in two ways to have five of each colour. 7 and 1; and 6 and 2 contradict the premise there are five of each colour. 5 and 3 leaves three of one colour alone. 4 and 4 has no way of combining to make five.
A solution for three is three black-and-white, 2 black-white-and-tan, and 3 tan-only. The colours can be rearranged to give a total of six solutions using this approach.