Number of Terms

First of all we write its general term :(Using Multinomial)

$\frac { 100! }{ { r }_{ 1 }!{ r }!_{ 2 }{ r }_{ 3 }! } \cdot { \left( { x }^{ 3 } \right) }^{ { r }_{ 1 } }{ \left( 1 \right) }^{ { r }_{ 2 } }{ \left( \frac { 1 }{ { x }^{ 3 } } \right) }^{ { r }_{ 3 } }$

where ${r}_1,{r}_2,{r}_3$ are integers and ${ r }_{ 1 }{ +r }_{ 2 }+{ r }_{ 3 }=100$ . On simplification ;

$\frac { 100! }{ { r }_{ 1 }!{ r }!_{ 2 }{ r }_{ 3 }! } \cdot { \left( x \right) }^{ 3\left( { r }_{ 1 }-{ r }_{ 3 } \right) }$ .

Now,since ${ r }_{ 1 }$ and ${ r }_{3}$ are integers $({ r }_{ 1 }-{ r }_{ 3 } )$ will also be an integer (say $r$ ) where $r\in \left[ -100,100 \right]$ Now distinct terms means terms having different powers of $x$ because similar powers of $x$ will add up together .So we will have powers of $x$ ranging from ${ x }^{ -300 },{ x }^{ -297 },{ x }^{ -294 }..$ to $..{ x }^{ -3 },{ x }^{ 0 },{ x }^{ 3 }..$ to $..{ x }^{ 294 },{ x }^{ 297 },{ x }^{ 300 }$ which are $\boxed { 201 }$ terms.