In the following sentence on idiomatic usage and word choice, the underlined expression needs improvement. Choose the option that best replaces the underlined words, or select “No improvement” if the sentence is already correct. His appointment as coach is yet another feather in his wing.

Introduction / Context:
This question tests understanding of a common English idiom that is frequently used in competitive examinations and formal writing. The phrase in focus is feather in his wing, which appears in the sentence about someone's appointment as a coach. The learner has to recognise which idiomatic expression is standard and natural in English, and which forms are incorrect or awkward.

Given Data / Assumptions:

• The sentence is about a person who has been appointed as a coach.
• The underlined part suggests a positive achievement or honour.
• The expression feather in his wing appears in the original sentence.
• The options provide slightly different idiomatic variations involving feather, hat, cap, wing, or no improvement.

Concept / Approach:
English has a fixed idiom a feather in someone's cap. It means an achievement, honour, or success that adds to a person's reputation. Idioms usually have fixed wording. Changing one key word, such as cap to hat or wing, breaks the idiom and makes the phrase sound wrong to a fluent speaker. Therefore, the task is to recall the correct fixed idiom and eliminate the incorrect or non standard variants.

Step-by-Step Solution:
Step 1: Identify that the sentence describes a positive achievement, namely his appointment as coach. Step 2: Recall the idiom that describes an achievement added to a person's record, which is a feather in his cap. Step 3: Compare this standard form with the versions given in the options. Step 4: Notice that feather in his cap appears exactly in option c. Step 5: Recognise that expressions like feather in his hat or feather in his wing are not standard idioms in English.

Verification / Alternative check:
If you consult good dictionaries or usage guides, they all record the idiom as a feather in someone's cap. You are very unlikely to see feather in his wing or badge in his hat used in the same positive, idiomatic sense. In exam style questions that test idioms, the examiner expects the candidate to choose the exact textbook version of the idiom, not a loose variation. That confirms that option c is the only fully correct and idiomatic choice.

Why Other Options Are Wrong:
Option a (a new badge in his hat) does not correspond to any standard idiom, so it sounds invented and unnatural in this context. Option b (another feather in his hat) is close but still wrong because the idiom uses cap, not hat. Option d (another jewel in his wing) is completely non idiomatic and confusing, because wings are not associated with this expression. Option e (no improvement) is incorrect because the original phrase feather in his wing is not a recognised idiom in English.

Common Pitfalls:
Learners often think that idioms are flexible, so they replace one noun with another similar noun such as cap and hat. However, idioms are fixed expressions, and even small changes can make them wrong. Another pitfall is reading too quickly and assuming that feather in his wing is correct because it looks similar to a known phrase. In exam conditions, it is important to slow down and match the idiom exactly with what you have seen in reliable study material.

Final Answer:
The standard and correct idiomatic expression is another feather in his cap, so option c is the best improvement.