Statistics Question involving Mode : Given the set of data $1,1,2,2,2,3,3,x,y$ .... I was studying for some quizzes when a wild question appears. It goes like this:

Given the set of data $1,1,2,2,2,3,3,x,y$, where $x$ and $y$ represent two different integers. If the mode is $2$, which of the following is true?
A. If $x = 1 \space or \space 2$, then $y = 3$
B. If $x = 1 \space or \space 3$, then $y = 2$
C. If $x = 1 \space or \space 2$, then $y = 2$
D. If $x = 1 \space or \space 3$, then $y = 3$

My work
The mode is the most common number occuring in a sequence of numbers. In this case, it's $2$.
If I think about A, that would be false, because if I'm going to put $x = 1$ and $y = 3$, completing the sequence $1,1,2,2,2,3,3,1,3$.
The mode will be $2$ and $3$. The problem states that mode must be $2$. If I'm going to put $x = 2$ and $y = 3$, completing the sequence $1,1,2,2,2,3,3,2,3$,
the mode will be $2$, making A correct. In short, A is half-truth
If I think about B, that would be true, because if I'm going to put $x = 1$ and $y = 2$, completing the sequence $1,1,2,2,2,3,3,1,2$.
The mode will be $2$. The problem states that mode must be $2$. If I'm going to put $x = 3$ and $y = 2$, completing the sequence $1,1,2,2,2,3,3,3,2$,
the mode will be $2$, making B correct. In short, B is true
If I think about C, that would be true, because if I'm going to put $x = 1$ and $y = 2$, completing the sequence $1,1,2,2,2,3,3,1,2$.
The mode will be $2$. The problem states that mode must be $2$. If I'm going to put $x = 2$ and $y = 2$, completing the sequence $1,1,2,2,2,3,3,2,2$,
the mode will be $2$, making C true. In short, C is true
If I think about D, that would be false, because if I'm going to put $x = 1$ and $y = 3$, completing the sequence $1,1,2,2,2,3,3,1,3$.
The mode will be $2$ and $3$. The problem states that mode must be $2$. If I'm going to put $x = 3$ and $y = 3$, completing the sequence $1,1,2,2,2,3,3,3,3$,
the mode will be $3$, making D false. In short, D is false
The problem is, I got two choices that says they were TRUE, which is the choices B and C.
I don't know what would be the correct one, akin to choosing two equally-potential suitors.
What might be the correct choice in this problem?

When you read something like: 

If $x=a$ or $b$, then $y=c$

What this means is that for the values of $x$, $y$ must equal $c$ in order to satisfy the condition that the sequence has a mode of 2. And if at least one value of $x$ does not satisfy the condition for $y$, then the entire thing is false.
For example, $A$ is not true. It's false - let's see why. 
As you correctly stated, we have sequences $$\begin{cases}1,1,2,2,2,3,3,1,y \\ 1,1,2,2,2,3,3,2,y\end{cases}$$
Now, for sequence 1, $y$ cannot be $3$, as this results in a mode other than $2$.
However, for sequence $2$, $y$ does not have to be $3$ - if $y=1,2$ or $3$ the mode will always be $2$. Thus, $y$ does not necessarily equal $3$. 
Now, when $x=2$ in this case, we are not guaranteed that $y=2$, hence $A$ is false.

Notice that for $B$, we have sequences $$\begin{cases} 1,1,2,2,2,3,3,1,y \\ 1,1,2,2,2,3,3,3,y\end{cases}$$
For sequence 1 $y$ must be $2$, or else the mode of the sequence is not $2$. 
For sequence 2, $y$ must be $2$ as well, or else the mode of the sequence does not equal $2$. 
Thus, due to the property that the sequence has a mode of $2$, when $x$ equals $1$ or $3$ we are guaranteed that $y$ must equal $2$. 
Hence $B$ is correct. 
