By drawing Pascal's Triangle showing same as a right angled triangle, and

applying Pascal's Identity and Pascal's Theorem, show that the three methods

may be used to determine equivalent values for the row of Pascal's triangle

containing the following binomial coefficients

$(\frac{15}{k}),0k15$

Explain the pattern of the odd numbers in the Pascal's Triangle from $($a$).$

Using a Large Language Model $($LLM$)($such as ChatGPT, Gemini, Claude,

etc.$),$ provide a solution for question $($b$)$ and describe the differences in the

two answers.