UGC NET CSE | October 2020 | Part 2 | Question: 2

Question

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?

• A. $4$
• B. $6$
• C. $7$
• D. $9$

Answer 1

6  indistinguishable objects into  4 indistinguishable boxes (direct example of rosen book)

The ways we can pack the books are

6  ,0,0,0  (only one way as 0,6,0,0  or 0,0,6,0  won’t be different  as boxes are identical)
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.

so total 9 ways hence ans is 4

Answer 2

Total ways = (Each box has at least 1 copy each) + (One box has 0 copy, Other 3 have at least 1 copy each) + (Two boxes have 0 copy, Other 2 have at least 1 copy each) + (Three boxes have 0 copy, and the last one has all 6 copies)

CASE-1: (Each box has at least 1 copy)

We are left with only 2 balls to distribute because we have distributed one ball to each of the boxes. Can be (1,1,0,0) or (2,0,0,0). [2 WAYS]

NOTE: (1,1,0,0) is same as (1,0,0,1), since boxes are identical

 

CASE-2: (One box has 0 copy, Other 3 have at least 1 copy each)

We are left with 3 balls to distribute. Can be (0,1,1,1) or (0,1,2,0) or (0,3,0,0). [3 WAYS]

 

CASE-3: (Two boxes have 0 copy, Other 2 have at least 1 copy each)

We are left with 4 balls to distribute. Can be (0,0,2,2) or (0,0,3,1) or (0,0,4,0). [3 WAYS]

 

CASE-4: (Three boxes have 0 copy, last one has all the copies)

Only one way (6,0,0,0) [1 WAY]

 

Therefore total =2+3+3+1=9 ways

If this comes as NAT type question, then remember this can’t be solved as “normal ball-bin problem” because here balls, as well as bins, are identical.