## Core Concept
The question tests the understanding of the association between Down's syndrome and neurological manifestations, particularly the development of speech difficulties. Down's syndrome, also known as Trisomy 21, is a genetic disorder caused by the presence of an extra copy of chromosome 21. It is associated with intellectual disability, delayed speech, and various physical anomalies.

## Why the Correct Answer is Right
Individuals with Down's syndrome are at an increased risk of developing Alzheimer's disease, which often presents with symptoms such as memory loss, confusion, and difficulty with speech. The patient's age (50 years) and the presence of Down's syndrome make Alzheimer's disease a likely diagnosis, especially since speech difficulty is a common symptom. Alzheimer's disease in Down's syndrome patients often starts at a younger age than in the general population.

## Why Each Wrong Option is Incorrect
- **Option A:** This option is not provided, but typically, incorrect options might include other conditions that cause speech difficulties, such as stroke or cerebral palsy, which are not as directly related to Down's syndrome and the age of the patient.
- **Option B:** Similarly, without the content, we can infer that any condition listed here would not be as strongly associated with both Down's syndrome and the specific symptom of speech difficulty as Alzheimer's disease.
- **Option C:** This would be another incorrect option, potentially including conditions like Parkinson's disease, which can cause speech difficulties but is less directly linked to Down's syndrome.

## Clinical Pearl / High-Yield Fact
A key point to remember is that individuals with Down's syndrome have a higher risk of developing Alzheimer's disease at a younger age than the general population. Speech difficulties, along with other cognitive impairments, can be an early manifestation of Alzheimer's disease in these patients. Therefore, any patient with Down's syndrome presenting with new cognitive or speech difficulties should be evaluated for Alzheimer's disease.

## Correct Answer Line
**Correct Answer: D. Alzheimer's disease**