From the given options, which word cannot be formed using the letters of the word PROXIMAL, assuming each letter can be used only as many times as it appears there?

Introduction / Context:
This is a classic word formation question. We are given the base word PROXIMAL and asked which of the listed words cannot be formed using its letters. The important condition, which is standard in such reasoning problems, is that each letter of the base word can be used at most as many times as it appears in the base word. The candidate must therefore carefully examine the letters in PROXIMAL and compare them with each option, checking for missing letters or excess usage of any letter.

Given Data / Assumptions:

• Base word: PROXIMAL.• Letters of PROXIMAL are: P, R, O, X, I, M, A, L.• Each of these letters appears exactly once in PROXIMAL.• Options: MOLAR, AXIOM, REALM, APRIL.• A word can be formed only if every letter in it exists in the base word with at least the same frequency.

Concept / Approach:
The strategy is straightforward. First, list the available letters from the base word, taking note that there is no repetition. Then, for each option, check letter by letter whether all letters are present in PROXIMAL and whether no letter appears more than once in the option. If any required letter is completely absent from PROXIMAL, that word cannot be formed. This approach is systematic and avoids guesswork.

Step-by-Step Solution:
Step 1: Note down the letters in PROXIMAL: {P, R, O, X, I, M, A, L}.Step 2: Examine option MOLAR: letters M, O, L, A, R.• M is in PROXIMAL.• O is in PROXIMAL.• L is in PROXIMAL.• A is in PROXIMAL.• R is in PROXIMAL.All letters are available and none is repeated more than once, so MOLAR can be formed.Step 3: Examine option AXIOM: letters A, X, I, O, M.• A, X, I, O and M all appear in PROXIMAL.Thus AXIOM can also be formed.Step 4: Examine option REALM: letters R, E, A, L, M.• R, A, L and M are present in PROXIMAL.• The letter E, however, does not appear anywhere in PROXIMAL.Therefore REALM cannot be formed from PROXIMAL because it requires the letter E.Step 5: Examine option APRIL: letters A, P, R, I, L.• A, P, R, I and L are all present in PROXIMAL.So APRIL can be formed.

Verification / Alternative check:
We can cross verify by writing PROXIMAL in sorted form as AILMOPRX and checking each option in sorted form. REALM becomes AELMR. Direct comparison shows that E is present in REALM but not in PROXIMAL. For the other options, every letter in their sorted form appears in AILMOPRX. Because letter availability is the only condition here and there are no repeated letters in the base word, this verification confirms that REALM is the only impossible word among the choices.

Why Other Options Are Wrong:
In this context, "wrong" means that the options can be formed and hence do not answer the question of which word cannot be formed. MOLAR uses exactly M, O, L, A and R, all of which are present in PROXIMAL. AXIOM requires A, X, I, O and M, again all present. APRIL uses A, P, R, I and L, all directly available from PROXIMAL. Since these words do not require any letter outside the set in PROXIMAL, they are all formable and therefore not the correct choice for this question.

Common Pitfalls:
One common mistake is to get distracted by the presence of unusual letters like X and assume that the hardest looking option must be the one that cannot be formed. Another is to forget that PROXIMAL contains no repeated letters, and therefore any option requiring a letter twice would also be impossible. In this particular case, the trap is subtler because all options use each letter at most once, and only close inspection reveals that REALM demands an E that does not exist in PROXIMAL. Carefully listing the available letters and checking each option letter by letter is the safest method.

Final Answer:
REALM