Find the missing numbers ($A$ and $B$) in the series and find the value of $B-A$:\n
$$51,57,63,A,75,81,B$$

Let's look at the difference between consecutive terms in the series:
$$57-51=+6$$

$$63-57=+6$$

Since the series follows a constant common difference pattern of $+6$:

$$A=63+6=69$$

Check the next terms: $69+6=75$, and $75+6=81$ (satisfies the series).

$$B=81+6=87$$

Now, calculate the required value of $B-A$:

$$B-A=87-69=18$$