The internal length, breadth and height of a cuboidal box are 12 cm, 10 cm and 8 cm, respectively. A cubical box has side (internal) 9 cm. Avin tries to pack 1800 cubical dice of side 1 cm each in these two boxes. The number of dice which are left unpacked is :

The correct option is 111.

To find the total number of dice that can be packed, we must first calculate the maximum number of dice that each box can hold. Since each die is a cube of side 1 cm, its volume is 1 cm³, meaning a box of volume V cm³ can hold exactly V dice (provided the dimensions are integers, which they are).

First, let us calculate the capacity of the cuboidal box.
The internal dimensions of the cuboidal box are:
Length = 12 cm, Breadth = 10 cm, Height = 8 cm.
The volume of this cuboidal box is given by:
$\mathrm{Volume}=\text{length}\times \text{breadth}\times \text{height}$
$\mathrm{Volume}=12\times 10\times 8=960{\text{cm}}^3$
Thus, the cuboidal box can hold exactly 960 dice.

Second, let us calculate the capacity of the cubical box.
The internal side length of the cubical box is 9 cm.
The volume of this cubical box is given by:
$\mathrm{Volume}={\text{side}}^3$
$\mathrm{Volume}=9^3=9\times 9\times 9=729{\text{cm}}^3$
Thus, the cubical box can hold exactly 729 dice.

Now, we find the total number of dice that can be packed in both boxes combined:
$\text{Total Capacity}=960+729=1689\text{dice}$

Avin wants to pack 1800 dice. The number of dice left unpacked is:
$\text{Dice Left Unpacked}=1800-1689=111$

Therefore, the number of dice which are left unpacked is 111.