Question

The first $3$ terms in the expansion of $(1+x+x^2{)}^{10}$ (in the ascending powers of $x$) are

• A. $1,10x,55x^2$
• B. $1,10x,65x^2$
• C. $1,10x,75x^2$
• D. $1,5x,10x^2$

Answer 1

The correct option is A $1,10x,55x^2$

GIven expansion is ${(1+x+x^2)}^{10}$,

We know that from multinomial theorem,

The general term in the expansion is $\frac{10!}{p!q!r!}{x}^{q+2r}$

where $p+q+r=10$,

For the first term consider $p=10,q=r=0$,

we get $1$,

Now for the coefficient of $x$ we consider $q+2r=1$,

$⟹p-r=9$,

As $p,q,r$ are natural numbers,for this condition to satisfy the only value of $r=0$,

$⟹p=9,q=1$,

$\therefore$ the second term is $10x$,

Now,to find the coefficient of $x^2$,

We consider the case of $q+2r=2$,

$⟹q=2,r=0$ and $q=0,r=1$,

$⟹p=8$ and$p=9$

$\therefore$ the third term is $45x^2+10x^2=55x^2$