Select the option in which the numbers share the same relationship as that shared by the given pair of numbers.
$$12:20$$

Let's find the mathematical relation for the pair $12:20$.
The numbers can be expressed in the form of $n(n+1)$ or $n^2+n$ for consecutive integers:

* For $12$: $3\times (3+1)=3\times 4=12$ (where $n=3$)
* For $20$: $4\times (4+1)=4\times 5=20$ (where $n=4$)

Thus, the pair consists of consecutive terms of the sequence defined by $n(n+1)$.

Let's test the options using the same logic:

*Option B ($30:42$):**
* For $30$: $5\times (5+1)=5\times 6=30$ (where $n=5$)
* For $42$: $6\times (6+1)=6\times 7=42$ (where $n=6$)

Since $30$ and $42$ are also consecutive terms of the same $n(n+1)$ pattern, option B shares the exact same relationship.