Question 1: The general expansion of \( (a+b)^{\wedge} n \) is given by: - A) \( a^{\wedge} n+b^{\wedge} n \) - B) \( \Sigma k=0^{\wedge} n n k a^{\wedge} n-k b^{\wedge} k \) - C) \( (a+b) \) - D) \( a+b \) Question 2 : In the binomial expansion of \( (x+2)^{\wedge} 3 \), what is the coefficient of \( x^{\wedge} 2 \) ? - A) 6 - B) 12 - C) 4 - D) 3 Question 3 : In the expansion

Sure, let's address each of your questions one by one. --- **Question 1:** *The general expansion of \( (a+b)^n \) is given by:* - **A)** \( a^n + b^n \) - **B)** \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \) - **C)** \( (a+b) \) - **D)** \( a+b \) **Answer:** **B)** The general expansion of \( (a+b)^n \) is given by the **Binomial Theorem**, which states: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] Here, \( \binom{n}{k} \) represents the binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \). --- **Question 2:** *In the binomial expansion of \( (x+2)^3 \), what is the coefficient of \( x^2 \)?* - **A)** 6 - **B)** 12 - **C)** 4 - **D)** 3 **Answer:** **A)** 6 To find the coefficient of \( x^2 \) in the expansion of \( (x + 2)^3 \), we can use the Binomial Theorem: \[ (x + 2)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} (2)^k \] The term containing \( x^2 \) corresponds to \( k = 1 \): \[ \binom{3}{1} x^{2} (2)^1 = 3 \times x^2 \times 2 = 6x^2 \] So, the coefficient of \( x^2 \) is **6**. --- **Question 3:** *It seems like your third question got cut off. Could you please provide the complete question so I can assist you effectively?* --- Feel free to ask if you have any more questions or need further clarification!