Let the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ represent three coterminous edges of a parallelopiped of volume $V$. Then the volume of the parallelopiped, whose coterminous edges are represented by $\overrightarrow{a}, \overrightarrow{b}+\overrightarrow{c}$ and $\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}$ is equal to

We know that,

Volume of parallelepiped whose edges determined by vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ is $\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]$.

Now it is given that $V$ is volume of parallelepiped $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$, so required volume of parallelepiped with new edges will be

$=\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}+\overrightarrow{c}& \overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}\end{array}\right]$

$=\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 1& 2& 3\end{array}\right|\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]$

$=\left\{\left(1\right)\left(3-2\right)-0+0\right\}\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]$

$=\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]$

$=V$

Hence this is the correct option.