Digits 0–9 are represented by letters a–j (0→a, 1→b, 2→c, 3→d, 4→e, 5→f, 6→g, 7→h, 8→i, 9→j). Evaluate dc × f − (bf − d) × d and express the final number back in letters.

Introduction / Context:
This item mixes symbolic digit naming with arithmetic. Each letter represents a digit via a fixed mapping. We must first translate letters to numbers, compute the result using standard arithmetic, then translate the final number back into letters.

Given Data / Assumptions:

• Mapping: a=0, b=1, c=2, d=3, e=4, f=5, g=6, h=7, i=8, j=9.
• Expression: dc × f − (bf − d) × d.
• Concatenations like dc and bf denote multi-digit numbers: dc = 32 and bf = 15.

Concept / Approach:
Translate, compute with ordinary arithmetic (respecting precedence with parentheses), then re-encode each digit of the final number using the same mapping.

Step-by-Step Solution:
1) dc = 32, f = 5, bf = 15, d = 3.2) Compute 32 × 5 = 160.3) Compute (15 − 3) × 3 = 12 × 3 = 36.4) Final: 160 − 36 = 124.5) Re-encode 1→b, 2→c, 4→e → "bce".

Verification / Alternative check:
Reverse-encoding "bce" → 1,2,4 to confirm the numeric result 124.

Why Other Options Are Wrong:

• abc (012), bcc (122), bcf (125): Do not match the computed value 124.

Common Pitfalls:
Treating dc or bf as products instead of concatenations, or forgetting the parentheses.

Final Answer:
bce