In the following question, out of the four alternatives, select the alternative which best expresses the meaning of the idiom / phrase "at somebody's expense".

Introduction / Context:
The idiom "at somebody's expense" is very common in everyday English, in both literal and figurative senses. Examinations often test this phrase because it appears in newspaper articles, conversations, and literature. Understanding its primary meaning helps you interpret sentences about who pays for something and who may be harmed or embarrassed by someone else's actions.

Given Data / Assumptions:

The idiom under test is: at somebody's expense.
We encounter it in contexts like "Dinner was at his expense" and "They made jokes at her expense".
The question asks for the alternative which best expresses the meaning of the idiom in general English usage.
We assume standard exam focus on the most straightforward meaning used in many sentences.

Concept / Approach:
The phrase at somebody's expense has two related meanings. The primary, literal meaning is "paid for by someone", as in a trip at the company's expense. The figurative meaning describes a situation where someone is harmed, embarrassed, or loses out while another benefits, as in a joke at someone's expense. In exam-style idiom questions, when you are given simple options, the safest and most widely applicable meaning is usually the literal one, especially when it matches common financial contexts such as travel, food, or facilities being paid for by someone else.

Step-by-Step Solution:
Step 1: Recall examples: "All costs were met at the government's expense" clearly means the government paid for everything. Step 2: Compare with option A: "Paid for by someone." This matches the literal meaning exactly and can be used in many such sentences. Step 3: Look at option D: "If one person benefits it is always at the loss of another." This reflects the figurative meaning of gaining at someone's cost, but the word "always" makes it too strong and not accurate in the purely financial sense. Step 4: Option B: "To blame a crime on someone else." describes framing someone, which is not the standard meaning of the idiom. Step 5: Option C: "The unlucky persons who lose a race." has nothing to do with paying for something or being harmed as such. Step 6: Option E: "To make a joke that embarrasses someone." is a specific figurative use (a joke at his expense), but the question asks for the best general meaning, which is broader and financial in many exam contexts.

Verification / Alternative check:
Consider the sentence: "The company organized a foreign tour at the CEO's expense." Here the only sensible meaning is that the CEO paid for it. If we substitute option A, we get "paid for by the CEO", which fits perfectly. If we try option E or D, the sentence becomes odd because a tour is not usually described mainly in terms of humour or loss. Similarly, in the sentence "He enjoys living at his parents' expense," the phrase clearly refers to money and material support, not just emotional harm. This strongly supports option A as the primary exam answer.

Why Other Options Are Wrong:
Option B describes shifting blame in a criminal context and is unrelated to payment or support.
Option C limits the idea to losing a race, which is far too narrow and unrelated to the idiom's established meanings.
Option D focuses on the idea that whenever one person benefits another must lose, which is not always true for this phrase and makes it too absolute.
Option E captures one specific figurative use (jokes at someone's expense) but does not cover the many sentences where the phrase simply refers to who is paying.

Common Pitfalls:
Students sometimes get attracted to dramatic-sounding options that highlight loss or humiliation, especially when they have seen the idiom used in contexts like "He cracked a joke at my expense." While that figurative use is correct, competitive exams often expect the broader and simpler interpretation first. A good strategy is to ask: can this meaning fit both money-related and harm-related contexts? "Paid for by someone" works well with travel, meals, and living arrangements, while the other options are too narrow or specific.

Final Answer:
The best expression of the idiom "at somebody's expense" is "paid for by someone".