Question

The number of ways in which the letters of the word PERSON can be placed in the squares of the given figure so that no row remains empty is

• A. $24\times 6!$
• B. $26\times 6!$
• C. $26\times 7!$
• D. $27\times 6!$

Answer 1

The correct option is B $26\times 6!$

There are $6$ different letters. we have to select $6$ squares, taking at least one from each row and then arranging in each selection. Let us first select places in each row such that no row remains empety.
$\begin{array}{cccc}{R}_1& R_2& R_3& \text{Number of selections}\\ 1& 1& 4& {}^2{C}_1\times {}^2{C}_1\times {}^4{C}_4=4\\ 1& 2& 3& {}^2{C}_1\times {}^2{C}_2\times {}^4{C}_3=8\\ 2& 1& 3& {}^2{C}_2\times {}^2{C}_1\times {}^4{C}_3=8\\ 2& 2& 2& {}^2{C}_2\times {}^2{C}_2\times {}^4{C}_2=6\end{array}$

$\therefore$ The total number of selections of $6$ squares is $4+8+8+6=26$. For each selection of $6$ squares, the number of arrangements of $6$ letters is $6!=720$.
Hence, the required number of ways is $26\times 720=18720$