If the first profit/loss is $x\%$ and second profit/loss is $y\%$ then overall profit/loss $\%$ is calculated as:
$\text{Total profit/loss=x+y+$\frac{xy}{100}\%$}$
- we use $+$ sign if profit happens
- we use $-$ sign if loss happens
It is given that the first $10\%$ profit is followed by $10\%$ loss so overall ;
Total profit/loss=$10+(-10)+\frac{10*(-10)}{100}\%\implies\frac{-100}{100}\%=-1\%$
we know that the minus sign indicates loss so overall $1\%$ loss made by the person.
Option $(C)$ is correct.