Symbiosis National Aptitude Test SNAP 2016 Paper

If there is only one box containing black ball, the boxes can be filled in 6 ways.
If there are two boxes containing black ball, the boxes can be filled in 5 ways. (The two black balls can be in either of the boxes (1,2), (2,3), (3,4), (4,5) or (5,6)).
If there are 3 boxes containing black ball the boxes can be filled in 4 ways viz.(123), (234), (345), (456).
Similarly if there are 4 boxes, it can be done in 3 ways viz.(1234), (2345), (3456), if there are 5 boxes it can be done in 2 ways viz.(12345), (23456) and all 6 boxes can have a black ball only in 1 way.
Hence, total number of ways = 6 + 5 + 4 + 3 + 2 + 1 = 21.