If a soccer ball can be painted with three different colors, how many unique ways can the ball be painted if each color must be used at least once?

Example Answer

There is not enough information for a single count. I need to know how many regions we paint, whether each region gets one color, and whether rotating the ball makes two patterns equivalent. Counting how many regions use each color does not count where those colors are placed.

As a clearly defined version, suppose there are n labeled panels, each panel gets one of three distinct colors, and every color must appear. Inclusion–exclusion gives 3 to the n minus 3 times 2 to the n plus 3. We start with all color assignments, subtract assignments missing each color, and add back the three assignments using just one color. For four labeled panels, that gives 81 minus 48 plus 3, or 36 patterns.

Why this works

It distinguishes color-frequency combinations from arrangements. The simplified model has explicit assumptions rather than pretending to solve an unspecified soccer-ball geometry.

Follow-up to prepare for

If rotations count as the same pattern, specify the panel geometry and use its rotation group, for example with Burnside’s lemma. The labeled-panel formula then cannot simply be reused.