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Plato b) There are five odd number boxes. If we go ahead and put one ball in to each of those five then distribute the remaining 25 balls into ten boxes in $\dbinom{25+10-1}{25}$ ways. Then odd boxes are nonempty.
c) This can be a trick thought experiment. Zero is an even number, so an empty box contains an even number of balls. Now think if "gluing" the balls into pairs of two each. Now put these 15 identical objects into the ten boxes.
That can be done in $\dbinom{15+10-1}{15}$ ways. Each box contains an number of the original balls.