Puzzle-20 (IIT-1978)

Six X’s have to be placed in the squares of the following figure, such that each row contains at least one X. In how many different ways can this be done?

Solution follows here:

Solution:

Number of positions = 2+4+2 = 8

Number of all possibilities = 6 X’s in 8 positions = 8C6 = 8C2 = 8*7/2 = 28

The condition is: “each row contains at least one X”

Out of all 28, the two exclusions are:

4 places in the middle row are filled with 4 X’s, remaining 2 X’s are filled in the first row, leaving the last row vacant.

4 places in the middle row are filled with 4 X’s, remaining 2 X’s are filled in the last row, leaving the first row vacant.

Barring these two, remaining 26 are the required ways.

Answer - 26