Question 6
A student divides all $30$ marbles among five boxes labeled $1,2,3,4,5$; empty boxes are allowed.
a) How many divisions are there, where two divisions differ if some box has a different number of marbles?
b) The student then paints every marble with one color. A color may be used on several marbles, but no two marbles in one box have the same color, and from any two boxes one cannot select eight marbles painted in four colors. Prove that at least ten colors are required, regardless of the division.
c) Exhibit a division and coloring satisfying b) with exactly ten colors.
a) How many divisions are there, where two divisions differ if some box has a different number of marbles?
b) The student then paints every marble with one color. A color may be used on several marbles, but no two marbles in one box have the same color, and from any two boxes one cannot select eight marbles painted in four colors. Prove that at least ten colors are required, regardless of the division.
c) Exhibit a division and coloring satisfying b) with exactly ten colors.