Let $\mathbb{F}_{11}$ be the finite field of 11 elements. Let $C=\{(x_1,x_2,\cdots, x_{10})~:~\sum_{i=1}^{10} x_{i}\equiv 0~(\text{mod}11),~\sum_{i=1}^{10} ix_{i}\equiv 0~(\text{mod}11),~x_{i}\in\mathbb{F}_{11}~\text{and}~x_{i}\neq 10 ~ \text{for all}~ 1\leq i\leq 10 \}$. Then find the cardinality of $C$.
Using properties of equivalent Codes, the above problem can be converted into the following equivalent form:
$C=\{(x_1,x_2,\cdots,x_{8},2x_{1}+3x_{2}+\cdots+9x_{8},8x_{1}+7x_{2}+\cdots+x_{8})\}$, where $x_1,x_2,\cdots,x_{8}$ run over the values $0,1,2,\dots,9$ and those values of $x_1,x_2,\cdots,x_{8}$ are omitted which give the digit '10' in either of the last two coordinate places.
Thanks in advance for any help, please.