In how many ways can the letters of the english alphabet be arranged s

Correct. 18 positions for the pair (A,B), and 2! ways to arrange them in those positions, and 24! ways to put the remaining 24.

A slightly different approach, just to confirm your answer:

• Choose a place for A and put B appropriately ahead: $26-(7+1)$ options
• Reorder A and B in every possible way: $2!$ options
• Choose $7$ letters out of the letters between C and Z: $\binom{26-2}{7}$
• Reorder those $7$ letters in every possible way: $7!$ options
• Reorder the remaining letters in every possible way: $(26-2-7)!$ options

The answer is therefore: $(26-(7+1))\cdot2!\cdot\binom{26-2}{7}\cdot7!\cdot(26-2-7)!=24!\cdot36$