The letters ABCDEFGH are to be used to form strings of length four 
The letters ABCDEFGH are to be used to form strings of length four.
  How many strings contain the letter A if repetitions are not allowed?

The answer that I have is :
$$ \frac{n!}{(n-r)!} - \frac{(n-1)!}{(n-r)!} = \frac{8!}{4!} - \frac{7!}{4!} = 8 \times 7 \times 6 \times 5 - (7 \times 6 \times 5) = 1470 $$ strings.
If you could confirm this for me or kindly guide in me the right direction, please do let me know.

I presume I am correct. Here is a detailed proof. 
First exclude 'A' and permute the rest (7P3). Which can be done in $\frac{7!}{4!}$ ways. 
Then, include 'A' back into those permuted cases. $|X_1|X_2|X_3|$ and as indicated by the vertical lines can be in 4 locations. So, the answer is 
$$\frac{7!}{4!} \times 4 = 840$$
