The patient has UTI secondary to E. coli. The organism is resistant to ampicillin and sensitive to cephalosporins. I think alot of people are confused over why penicillin binding protein alteration wasn't the correct answer so here is my take:

Ampicillin - Penicillin antibiotic, with a beta lactam ring.

Mechanisms of resistance - Penicillinase (Beta lactamase which cleaves the beta lactam ring of penicillin drugs making them ineffective) OR PBP alteration (penicillin binding protein alteration).

Ceftriaxone - third generation Cephalosporin. Mechanisms of resistance - CEPHALOSPORINASE (Another type of beta lactamase), OR PBP alteration.

The question asks, which mechanism of resistance does the organism produce that causes ampicillin to be inactivated BUT ceftriaxone to be resistant.

It's the TYPE of beta lactamase produced by the two drugs that are different. so Beta lactamase production against Ampicillin (Penicillinase) would inactivate Ampicillin but NOT Ceftriaxone (cephalospoirinase).

If the mechanism of resistance were penicillin binding protein alteration, BOTH drugs would be resistant, because that mechanism of resistance is SHARED by both drugs.

It's a shitty question, which is nitpicky and goes after a pretty dumb detail.

Ampicillin (and amoxicillin and aminopenicillins) are penicillinase-sensitive penicilins. Resistance occurs when bacteria produce penicillinase (a type of beta lactamase) which cleaves the B-lactam ring. Resistance to cephalosporins occurs when bacteria produce cephalosporins (a type of beta lactamase) OR through structural changes in penicillin-binding proteins (transpeptidases). FA2018 188-189

Though ampicillin and ceftriaxone are both considered beta lactam antibiotics, Most cephalosporins have intrinsic resistance to the beta-lactamases that inactivate other beta-lactams (FA2020 p189). Cephalospoin resistance is caused by structural changes in the transpeptidases (also called penicillin binding protein), or inactivation by cephalosporinases (a specific type of beta-lactamase)