Question 65
A student divides $30$ marbles into $5$ boxes labelled $1, 2, 3, 4, 5$ (there may be a box without marble).
Part (a). How many ways are there to divide marbles into boxes (two ways are different if there is a box with different number of marbles)?
Part (b). After dividing, this student paints those marbles by a number of colors (each marble have one color, one color can be painted for many marbles), such that there does not exist $2$ marbles in the same box, having a mutual 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 at least $10$ colors to paint the marbles.
Part (c). Find a division so that the student can use exactly $10$ colors to paint the marbles that satisfies the conditions in question b).
Part (a). How many ways are there to divide marbles into boxes (two ways are different if there is a box with different number of marbles)?
Part (b). After dividing, this student paints those marbles by a number of colors (each marble have one color, one color can be painted for many marbles), such that there does not exist $2$ marbles in the same box, having a mutual 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 at least $10$ colors to paint the marbles.
Part (c). Find a division so that the student can use exactly $10$ colors to paint the marbles that satisfies the conditions in question b).