The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the…

Question

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :

Accepted Answer

Let $x,y,z$ be the number of box which are filled

$\Rightarrow 1\le x\le 3,1\le y\le 3,1\le z\le 2$

$x$
 $y$
 $z$
 Number of ways

3
 1
 1
 ${}^{3}{C}_{3}\cdot {}^{3}{C}_{1}\cdot {}^{2}{C}_{1}=6$

2
 2
 1
 ${}^{3}{C}_{2}\cdot {}^{3}{C}_{2}\cdot {}^{2}{C}_{1}=18$

1
 3
 1
 ${}^{3}{C}_{1}\cdot {}^{3}{C}_{3}\cdot {}^{2}{C}_{1}=6$

2
 1
 2
 ${}^{3}{C}_{2}\cdot {}^{3}{C}_{1}\cdot {}^{2}{C}_{2}=9$

1
 2
 2
 ${}^{3}{C}_{1}\cdot {}^{3}{C}_{2}\cdot {}^{2}{C}_{2}=9$

Total ways $=(48)$ to fill boxes

Now to arrange $a,b,c,d$ and $e$

Number of ways will be 48.5! = 5760

Solutions · 1