Let's first give pens to all of them to satisfy their minimum requirement.
$$\begin{array}{} \underbrace{\text{Amal}}_{1(\geq 1)} & \underbrace{\text{Bimal}}_{2(\geq 2)} & \underbrace{\text{Kamal}}_{3(\geq 3)} \end{array}$$
After that, pens left $ = 8-(1+2+3) = 8-6 = 2$ pens.
These $2$ pens can be distributed among Amal, Bimal, and Kamal in the following ways:
$$\begin{array}{ccc} \underline{\text{Amal}} & \underline{\text{Bimal}} & \underline{\text{Kamal}} \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \\ 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}$$
$\therefore$ The total number of distribution possible $=6$ ways.
$\textbf{Second Method:}$
Let's,
- Amal $=a$
- Bimal $=b$
- Kamal $=c$
$a+b+c=8; a\geq1, b\geq2, c\geq3$
$\Rightarrow a+b+c=8; a-1\geq 0, b-2\geq 0, c-3\geq 0 \quad \longrightarrow(1)$
Let,
- $a-1=x \implies a=x+1$
- $b-2=y \implies b=y+2$
- $c-3=z \implies c=z+3$
Now, $x+1+y+2+z+3=8; x\geq 0, y\geq 0, z\geq 0$
$\Rightarrow x+y+z= 8-(1+2+3); x\geq 0, y\geq 0, z\geq 0$
$\Rightarrow x+y+z= 2; x\geq 0, y\geq 0, z \geq 0$
Number of distribution possible $ = \;^{3+2-1}C_2 = \; ^4C_2 = \frac{4!}{2!2!} = \frac{4\times3\times2!}{2\times1\times2!}=6\;\text{ways}.$
Reference: https://brilliant.org/wiki/integer-equations-star-and-bars/
Correct Answer $:\text{B}$