Question

Six $Xs$ have to be placed in the squares of figure below in such a way that each row contains at least one $X$. In how many different ways can this be done?

Answer 1

As all the X's are identical, the question is of selection of 6 squares from 8 squares, so that no row remains empty. Here $R_1$ has 2 squares, $R_2$ has 4 squares, and $R_3$ has 2 squares. The selection scheme is as follows:

$R_1$	$R_2$	$R_3$
1	4	1
1	3	2
2	3	1
2	2	2

Therefore, number of selection is
${}^2{C}_1{\times }^4{C}_4{\times }^2{C}_1{+}^2{C}_1{\times }^4{C}_3{\times }^2{C}_2{+}^2{C}_2{\times }^4{C}_3{\times }^2{C}_1{+}^2{C}_2{\times }^4{C}_2{\times }^2{C}_2=4+8+8+6=26$