A man has 20 equal marbles. He need to put all of them into 4 different boxes. How many ways are there to put the marbles into the box ? ( He may leave some boxes empty )
1)23*22*21/3*2*1
2)20*19*18*17/4*3*2*1
3)20*19*18/3*2*1
It will be easy for me ,if the question is "A man has 20 equal marbles. He need to put all of them into 4 different boxes. How many ways are there to put the marbles into the box ?".But it is given that he may leave some boxes empty.
Hint: Suppose you have n distinguishable boxes and k marbles. And a box can have no marbles up to k marbles. Then the number of ways are there to put the marbles into the box is
$$\binom{k+n-1}{k}$$
Hint : The answer you are looking for is given by the partition funtion $p(n,k)$ which counts the number of ways to write the integer $n$ as a sum of $k$ non-negative integers.
(also you can try to work it out rigorously: The number of ways to place 20 objects into 2 boxes is 21. $(20+0, 19+1, 18+2 , \ldots , 1+19, 0+20)$. Add another box and work out the numbers. )
