Step 1: Understand the term 'helminths'. Helminths are parasitic worms, which include flatworms (like flukes and tapeworms) and roundworms (nematodes). They are multicellular organisms that infect hosts and cause various diseases.
Step 2: Identify each organism listed in the problem. Schistosoma mansoni is a type of blood fluke (a flatworm), Taenia solium is a tapeworm (a flatworm), and Ascaris lumbricoides is a roundworm (a nematode). All three are helminths.
Step 3: Recognize the exception. Escherichia coli is a bacterium, a single-celled prokaryotic microorganism, not a worm or helminth.
Step 4: Conclude that since Escherichia coli is a bacterium and not a worm, it is the correct answer to the question asking for the organism that is NOT a helminth.
Step 5: Summarize that the key to solving this problem is knowing the classification of each organism and understanding that helminths are multicellular parasitic worms, whereas bacteria like Escherichia coli are not.