Question

The constant term in the expansion of \( \left(1+\frac{1}{x}\right)^{20} \left(30x(1+x)^{29} + (1+x)^{30}\right) \) is

• A. \( {}^{50}C_{20} + 30 \cdot {}^{50}C_{29} \)
• B. \( {}^{50}C_{19} + 30 \cdot {}^{49}C_{19} \)
• C. \( {}^{50}C_{20} + 30 \cdot {}^{49}C_{20} \)
• D. \( {}^{50}C_{20} + 30 \cdot {}^{49}C_{19} \)

Hint:

To find the coefficient of \( x^k \) in \( x^m (1+x)^n \), you simply need to find the coefficient of \( x^{k-m} \) in the binomial expansion of \( (1+x)^n \), which is \( {}^{n}C_{k-m} \). Be careful with negative powers.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:

We need to simplify the given expression into a form where we can apply the binomial theorem to find the coefficient of \( x^0 \) (the constant term).
Step 2: Detailed Explanation:

Let the expression be \( E \). \[ E = \left(1+\frac{1}{x}\right)^{20} \left[ 30x(1+x)^{29} + (1+x)^{30} \right] \] Simplify the first term: \[ \left(1+\frac{1}{x}\right)^{20} = \left(\frac{x+1}{x}\right)^{20} = x^{-20}(1+x)^{20} \] Substitute this back into \( E \): \[ E = x^{-20}(1+x)^{20} \left[ 30x(1+x)^{29} + (1+x)^{30} \right] \] Distribute the outer term: \[ E = 30x \cdot x^{-20}(1+x)^{20}(1+x)^{29} + x^{-20}(1+x)^{20}(1+x)^{30} \] \[ E = 30x^{-19}(1+x)^{49} + x^{-20}(1+x)^{50} \] We need the constant term, i.e., the coefficient of \( x^0 \). Let's analyze the two parts separately: 1. First Part: \( 30x^{-19}(1+x)^{49} \) To get \( x^0 \), we need the term involving \( x^{19} \) from the expansion of \( (1+x)^{49} \), because \( x^{-19} \cdot x^{19} = x^0 \). The coefficient of \( x^{19} \) in \( (1+x)^{49} \) is \( {}^{49}C_{19} \). So, the constant term from this part is \( 30 \cdot {}^{49}C_{19} \). 2. Second Part: \( x^{-20}(1+x)^{50} \) To get \( x^0 \), we need the term involving \( x^{20} \) from the expansion of \( (1+x)^{50} \), because \( x^{-20} \cdot x^{20} = x^0 \). The coefficient of \( x^{20} \) in \( (1+x)^{50} \) is \( {}^{50}C_{20} \). So, the constant term from this part is \( {}^{50}C_{20} \). Total constant term: \[ C = {}^{50}C_{20} + 30 \cdot {}^{49}C_{19} \]
Step 4: Final Answer:

The correct option is (D).