Question

The coefficient of \( x^5 \) in the expansion of \( (1+x^2)^5(1+x)^4 \) is:

• A. \( 30 \)
• B. \( 60 \)
• C. \( 40 \)
• D. \( 10 \)
• E. \( 45 \)

Hint:

Always list out the available powers of \( x \) for both binomials first to avoid checking invalid combinations.

Answer 1

The Correct Option is B

Concept: The coefficient of a specific power in a product of two binomial expansions is the sum of products of terms from each expansion whose powers add up to the target power.

Step 1: Analyzing the general terms.
From \( (1+x^2)^5 \), the powers of \( x \) are \( 0, 2, 4 \dots \). From \( (1+x)^4 \), the powers of \( x \) are \( 0, 1, 2, 3, 4 \). We need \( x^{p} \cdot x^{q} = x^5 \), where \( p \in \{0, 2, 4\} \) and \( q \in \{0, 1, 2, 3, 4\} \).

Step 2: Identifying valid combinations.
1. \( p=2, q=3 \): \( \binom{5}{1} \cdot \binom{4}{3} = 5 \cdot 4 = 20 \) 2. \( p=4, q=1 \): \( \binom{5}{2} \cdot \binom{4}{1} = 10 \cdot 4 = 40 \) Total coefficient = \( 20 + 40 = 60 \).