Indistinguishable objects and Indistinguishable boxes

Question

Can please anybody explain how to solve the case for indistinguishable objects and indistinguishable boxes.

Ques. How many ways are there to pack six copies of the same book into four identical boxes,where a box can contain as many as six books?

                       -- question from Rosen

In Rosen it's said that it's equivalent of writing n as the sum of at most k positive integers in non increasing order. How??

Comments

Is the answer (4 + 6 -1) C 6?

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No sir there are 9 possible ways to do

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it's equivalent of writing n as the sum of at most k positive integers in non increasing order.

Na462 if we exclude above line from question then we have 126 ways using   smsubham formula.Am I right?

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create cases:

case 1: ONLY 1 BOX USED → all the 6 books in one box (6) = 1 way

case 2: ONLY 2 BOXES USED → (1,5),(2,4), (3,3) = 3 ways

NOTE: DON’T ARRANGE (don’t multiply 3 with 2! ) COZ BOXES AND BOOKS ARE IDENTICAL i.e. (5,1) <==> (1,5)

case 3: ONLY 3 BOXES USED

and so on… till

case 6: ALL 6 BOXES USED → (1)(1)(1)(1)(1)(1) = 1way

answer will be 9 ways.

Answer 1

Here n=6 and k=4
Write 6 as the sum of at most 4 positive integers in non-increasing order.
We can do it in this way- 
6+0+0+0  // One box contains 6 copies of the book and other boxes are empty
5+1+0+0
4+2+0+0
4+1+1+0
3+3+0+0
3+2+1+0
3+1+1+1
2+2+2+0
2+2+1+1
Total number of possible ways = 9
Reference - Check This

Comments

Nicely Explained