In how many ways can 7 marbles be placed in 10 boxes if a th

In how many ways can 7 marbles be placed in 10 boxes if:

(a) the marbles are different colors, and marbles can share a box;

(b) the marbles are different colors, and marbles cannot share a box;

(c) the marbles are all the same color, and marbles cannot share a box;

(d) the marbles are all the same color, and marbles can share a box

Please show work Thank You

Solution

a)

The first marble can be placed in one of the 10 boxes in 10 ways.

The second marble can be placed in one of the 10 boxes in 10 ways.

The third marble can be placed in one of the 10 boxes in 10 waysand so on.

Hence the total no of ways = 10⁷ =10000000

b)

Let us select the boxes in which the marbles needed to be placed.

There are 10 boxes in which 7 boxes are to be selected. this can be done in ¹⁰C₇ ways.

Now there are 7 boxes in which 7 different colured marbes need to be placed. this can be done in 7! ways

Hence total no of ways =¹⁰C₇*7! =604800 ways.

c)

Let us select the boxes in which the marbles needed to be placed.

There are 10 boxes in which 7 boxes are to be selected. this can be done in ¹⁰C₇ ways.

Now there are 7 boxes in which 7 same colured marbes need to be placed. this can be done in 7! ways

Hence total no of ways =¹⁰C₇*7! =604800 ways.

d)

The first marble can be placed in one of the 10 boxes in 10 ways.

The second marble can be placed in one of the 10 boxes in 10 ways.

The third marble can be placed in one of the 10 boxes in 10 waysand so on.

Hence the total no of ways = 10⁷ =10000000