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  4. Six X 's have to be placed in the square of the figure such that each row contains atleast one ' X '. In how many different ways can this be done?

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Q. Six $X$ 's have to be placed in the square of the figure such that each row contains atleast one ' $X$ '. In how many different ways can this be done?Mathematics Question Image

Permutations and Combinations

Solution:

In all, we have 8 squares in which 6 ' $X$ ' have to be placed and it can be done in ${ }^8 C_6=28$ ways.
But this includes the possibility that either the top or horizontal row does not have any ' $X$ '. Since, we want each row must have atleast one ' $X$ ', these two possibilities are to be excluded.
Hence, required number of ways $=28-2=26$