Question

Find the number of ways in which $AAABBB$ can be placed in the squares of Fig as shown, so that no row remains empty.

Answer 1

2nd and 4th row block has to be selected.
One row out of 1st, 3rd, and 5th will contain two letters and remaining one one letter.
Selection of one row that contains two letters is done in ${}^3{C}_1$ ways.
Now two block can be selected from this row in ${}^3{C}_2$ ways.
From each of remaining two rows, one blocks can be selected in ${}^3{C}_1$ ways.
Hence, total selections ${=}^3{C}_1{\cdot }^3{C}_2{\cdot }^3{C}_1{\cdot }^3{C}_1=81$
Number of ways of filling A, A, A, B, B, B$=81\times \frac{6!}{3!3!}$
$=81\times 20=1620$