Question

Six $P^{\prime }s$ have to be placed in the squares of the diagram given below such that each column contains at least one $P$. In how many ways this can be done?

Answer 1

$C_1$	$C_2$	$C_3$
$2$	$3$	$1$
$2$	$2$	$2$
$1$	$4$	$1$
$1$	$3$	$2$

$C1\to$First column

$C2\to$Second column

$C3\to$Third column

Hence, the required number of ways

${=}^2{C}_2{\times }^4{C}_3{\times }^2{C}_1{+}^2{C}_2{\times }^4{C}_2{\times }^2{C}_2{+}^2{C}_1{\times }^4{C}_4{\times }^2{C}_1$${+}^2{C}_1{\times }^4{C}_3{\times }^2{C}_2$

$=1\times 4\times 2+1\times 6\times 1+2\times 1\times 2+2\times 4\times 1$

$=8+6+4+8=26$