Question

The number of ways in which $5X^{\prime }s$ can be placed in the square of the figure so that no row remains empty:

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

Answer 1

Answer: (B)

$44$

Case $-\left(1\right):$

let number of $X^{\prime }s$ in first row be $3$,second row be $1$,third row be $1$

no of ways for this is ${}^4{C}_3{\times }^2{C}_1{\times }^2{C}_1=16$

Case $-\left(2\right):$

let number of $X^{\prime }s$ in first row be $2$,second row be $1$ or $2$,third row be $2$ or $1$ respectively

no of ways for this is $2{(}^4{C}_2{\times }^2{C}_1{\times }^2{C}_1)=24$

Case $-\left(3\right):$

let number of $X^{\prime }s$ in first row be $1$,second row be $2$,third row be $2$

no of ways for this is ${}^4{C}_1{\times }^2{C}_2{\times }^2{C}_2=4$

Total number of ways is $16+24+4=44$