I have a big doubt
In $(a+b+c)^{20}$ we take b+c as a single term and a as a single term and then we expand and then we find the number of terms
which comes out to be $1+2......+21=21*22/2=231$
But in $(1+x+x^2)^{20}$ we take the number of terms as the highest power+1=41. Then why not the same in $(a+b+c)^{20}$. Why number of terms in this case is $231$ and not $21$? How should we know that in which question we should club two terms and in which question we should look simply at the powers and tell about the number of terms?
This is because in the expansion of $$(a+b+c)^{20}$$ the variables $a,b,c$ cannot be combined, whereas some terms in $$(1+x+x^2)^{20}$$ can be combined.
Let's first consider a simpler example: $$(a+b+c)^2$$ compared to $$(1+x+x^2)^2$$ when we expand the first one, we get $$a^2+b^2+c^2+2ab+2bc+2ac$$ However, in the second one, $a=1$, $b=x$, and $c=x^2$, and so the terms for $b^2=x^2$ and $ac=x^2$ can be combined, whereas they cannot be combined if $a,b,c$ are left as ambiguous variables.