Question

The coefficient of \( x^{20} \) in the expansion of \( (1 + 3x + 3x^2 + x^3)^{20} \):

• A. \( \binom{60}{40} \)
• B. \( \binom{30}{20} \)
• C. \( \binom{15}{2} \)
• D. None of these

Hint:

Look for pattern simplification or treat as multinomial to find term matching required power.

Answer 1

The Correct Option is A

We are given the expansion: \[ (1 + 3x + 3x^2 + x^3)^{20} \] Notice that: \[ 1 + 3x + 3x^2 + x^3 = (1 + x)^3 \] Hence, the given expression simplifies as: \[ [(1 + x)^3]^{20} = (1 + x)^{60} \] Now, we are to find the coefficient of \( x^{20} \) in the expansion of \( (1 + x)^{60} \).
From the binomial expansion: \[ (1 + x)^{60} = \sum_{k=0}^{60} \binom{60}{k} x^k \] So, the coefficient of \( x^{20} \) is: \[ \binom{60}{20} \] Using the identity: \[ \binom{n}{r} = \binom{n}{n - r} \Rightarrow \binom{60}{20} = \binom{60}{40} \] Thus, the required coefficient is \( \binom{60}{40} \).