MathDB

A student divides all

30

marbles into

5

boxes numbered

1, 2, 3, 4, 5

(after being divided, there may be a box with no marbles). a) How many ways are there to divide marbles into boxes (are two different ways if there is a box with a different number of marbles)? b) After dividing, the student paints those

30

marbles by a number of colors (each with the same color, one color can be painted for many marbles), so that there are no

2

marbles in the same box. have the same color and from any

2

boxes it is impossible to choose

8

marbles painted in

4

colors. Prove that for every division, the student must use no less than

10

colors to paint the marbles. c) Show a division so that with exactly

10

colors the student can paint the marbles that satisfy the conditions in question b).

Problem Statement