Here are the solutions to each question:

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1. What is the coefficient of a specific term in the binomial expansion using Pascal's Triangle?

• The coefficient is determined by selecting the appropriate row and entry in Pascal's Triangle.
• Coefficients are $\binom{n}{r}$, where $n$ is the row (starting from 0) and $r$ is the position.
  Example: Row 3 has values $1, 3, 3, 1$.

Answer: d. 10

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2. In the binomial expansion of $(x+y)^n$, what is the coefficient of a term?

• Same logic as above: Locate the term using $\binom{n}{r}$.
  Answer: d. 10

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3. Sum of numbers in the third row of Pascal's Triangle:

• The sum of numbers in the $n$-th row is $2^n$.
• For row 3, the sum is $2^3 = 8$.
  Answer: d. 8

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4. Which row corresponds to specific coefficients?

• Expand $(x+y)^6$: coefficients are $1, 6, 15, 20, 15, 6, 1$.
  Answer: a. 1, 6, 15, 20, 15, 6, 1

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5. Binomial expansion of $(x+y)^n$:

• Pascal's Triangle provides the coefficients. Expand $(x+y)^n$ directly.
  Answer: Option dependent on specific term given

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6. What is the binomial expansion of a polynomial?

• Depends on the term; expand using coefficients from Pascal's Triangle.
  Answer: Option dependent on specific polynomial.

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7. Pattern in diagonals of Pascal's Triangle:

• First diagonal: all 1's.
• Second diagonal: consecutive integers.
• Third diagonal: triangular numbers.
• Fourth diagonal: tetrahedral numbers.
  Answer: b. Fibonacci sequence

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8. Coefficients of the expansion $(x+y)^3$:

• Coefficients are $1, 3, 3, 1$.
  Answer: a. 1, 3, 3, 1

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9. Which row adds to 16?

• Row sums are $2^n$. For $2^4 = 16$, row 4 sums to 16.
  Answer: b. Fourth row

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10. Which row sums to 128?

• Row sums are $2^n$. For $2^7 = 128$, row 7 sums to 128.
  Answer: b. Seventh row

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Follow-up Questions:

1. How is Pascal’s Triangle constructed step-by-step?
2. What is the relationship between Pascal’s Triangle and binomial expansions?
3. Can you explain the pattern of Fibonacci numbers in Pascal’s Triangle?
4. What is the role of combinations $\binom{n}{r}$ in Pascal’s Triangle?
5. How does Pascal’s Triangle connect to real-world applications, such as probability?

Tip:

To find the coefficients in $(x + y)^n$, look at the $n$-th row of Pascal's Triangle. Each entry corresponds to $\binom{n}{r}$.