Question

The number of ways in which 5 Xs can be placed in the squares of the figure so that no row remains empty is

• A. $97$
• B. $44$
• C. $100$
• D. $126$

Answer 1

The correct option is B $44$

We have $5x^{\prime }s$ which have to be filled in the $3$ rows so that no row is empty. there are $4$ possibilities :

$i)(3,1,1)$ $ii)(2,2,1)$ $iii)(2,1,2)$ $iv)(1,2,2)$

Where in $(x,y,z),$ $x=$ no. of $x^{\prime }s$ in $1st$ row, $y=$ no. of $x^{\prime }s$ in $2nd$ row, $z=$ no. of $x^{\prime }s$ in $3rd$ row

For Case $i).3x^{\prime }s$ can be placed in $4$ boxes in ${}^4{C}_3$ ways $=4$

$1x$ in $2^{nd}$ and $3^{rd}$ row can be placed in ${}^2{C}_1,{}^2{C}_1$ ways respectively.

Total $4\times 2\times 2=16$

Similarly,

For Case ii) Total $={}^4{C}_1\times {}^2{C}_2\times {}^2{C}_1=12$

Case iii) Total $={}^4{C}_2\times {}^2{C}_1\times {}^2{C}_2=12$

Case iv) Total $={}^4{C}_1\times {}^2{C}_2\times {}^2{C}_2=4$

$\Rightarrow$ Required sum $=16+12+12+4=44$

$\therefore$ Totally $44$ possible ways.

Hence, the answer is $44.$