Isabella has $6$ coupons that can be redeemed for free ice cream cones at Pete's Sweet Treats. In order to make the coupons last, she decides that she will redeem one every $10$ days until she has used them all. She knows that Pete's is closed on Sundays, but as she circles the $6$ dates on her calendar, she realizes that no circled date falls on a Sunday.  On what day of the week does Isabella redeem her first coupon?
$\textbf{(A) }\text{Monday}\qquad\textbf{(B) }\text{Tuesday}\qquad\textbf{(C) }\text{Wednesday}\qquad\textbf{(D) }\text{Thursday}\qquad\textbf{(E) }\text{Friday}$
Let $\text{Day }1$ to $\text{Day\\ }2$ denote a day where one coupon is redeemed and the day when the second coupon is redeemed. 
If she starts on a $\text{Monday}$, she redeems her next coupon on $\text{Thursday}$. 
$\text{Thursday}$ to $\text{Sunday}$.
Thus, $\textbf{(A)}\ \text{Monday}$ is incorrect.

If she starts on a $\text{Tuesday}$, she redeems her next coupon on $\text{Friday}$.
$\text{Friday}$ to $\text{Monday}$.
$\text{Monday}$ to $\text{Thursday}$.
$\text{Thursday}$ to $\text{Sunday}$.
Thus $\textbf{(B)}\ \text{Tuesday}$ is incorrect.

If she starts on a $\text{Wednesday}$, she redeems her next coupon on $\text{Saturday}$.
$\text{Saturday}$ to $\text{Tuesday}$.
$\text{Tuesday}$ to $\text{Friday}$.
$\text{Friday}$ to $\text{Monday}$.
$\text{Monday}$ to $\text{Thursday}$.
And on $\text{Thursday}$, she redeems her last coupon. 

No Sunday occured; thus, $\boxed{\textbf{(C)}\ \text{Wednesday}}$ is correct. 

Checking for the other options, 

If she starts on a $\text{Thursday}$, she redeems her next coupon on $\text{Sunday}$.
Thus, $\textbf{(D)}\ \text{Thursday}$ is incorrect.

If she starts on a $\text{Friday}$, she redeems her next coupon on $\text{Monday}$.
$\text{Monday}$ to $\text{Thursday}$.
$\text{Thursday}$ to $\text{Sunday}$.

Checking for the other options gave us negative results; thus, the answer is $\boxed{\textbf{(C)}\ \text{Wednesday}}$.