[MetsFan09]

In the attached question the answer is choice D through E1 elimination.  But I am wondering why can't choice B also be a choice through SN1?

[g-bones]

these are strongly basic conditions, are you sure it is through an E1 pathway?  you are right in that, if a carbocation is formed, than either hydroxide or ethanol/ethoxide could attack just as likely.  But these conditions are not conducive to carbocation formation.  strongly basic conditions favor E2 and SN2 pathways. you will notice that the halide is attached at a quaternary  center, which is too hindered for SN2. elimination to form the more substituted double bond is likely via E2.  hope this helps.     

[MetsFan09]

Yes that makes perfect sense I forgot that E2 mechanisms favor 3 degree while SN2 do not thank you.