Question

If the number of ways in which the letters of the word $TEACHER$ can be placed in the squares of the given figure so that no row remains empty is $\left(k\right)7!$, then find the value of $k$.

Answer 1

Total no. of ways $={}^{10}{C}_7\times \frac{7!}{2!}$

No of ways when rows are empty$=2$

$\left[\right(R_1,R_2,R_4)$ occupied or $(R_3,R_4)$ occupied$]$

$=2\times \frac{7!}{2!}$

Required ways $={}^{10}{C}_7\times \frac{7!}{2!}-2\times \frac{7!}{2!}$

$=\left(\frac{{}^{10}{C}_7-2}{2}\right)7!$

$=59\times 7!$

$\Rightarrow k=59$.