If $n$ is  an integer greater than $1$ , then
$a-{}^{n}C_1(a-1)+{}^{n}C_2(a-2)-\dots +(-1{)}^n(a-n)=$

Consider the following statements
$S_1$: The total number of terms in ${\left(x^2+2x+1\right)}^{10}$ is $21$.
${\mathrm{S}}_2$: The coefficient of $x^{10}$ in ${\left(x^2+\frac{1}{x}\right)}^{20}$ is ${}^{20}C_{10}$.

${\mathrm{S}}_3:$ The middle term in the expansion of $(1+x{)}^{12}$ is ${}^{12}C_6{x}^6$.

${\mathrm{S}}_4$: If the coefficients of the fifth and ninth term in the expansion of $(1+x{)}^n$ are same then $n=12$.
Now identify the correct combination of true statements.