Question

The coefficient of \( x^5 \) in the expansion of \( (x + 3)^8 \) is

• A. 1542
• B. 1512
• C. 2512
• D. 2542
• E. 2452

Hint:

To find coefficient of \( x^r \), equate power with \( n-k \) and solve for \( k \).

Answer 1

The Correct Option is B

Concept: Using the Binomial Theorem: \[ (x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k \] The general term is: \[ T_{k+1} = \binom{n}{k} x^{n-k} a^k \]

Step 1: Identify required term.
We need coefficient of \( x^5 \), so: \[ n - k = 5 \Rightarrow 8 - k = 5 \Rightarrow k = 3 \]

Step 2: Write the required term.
\[ T = \binom{8}{3} x^5 \cdot 3^3 \]

Step 3: Compute values.
\[ \binom{8}{3} = 56,\quad 3^3 = 27 \]

Step 4: Multiply.
\[ 56 \times 27 = 1512 \] Final Answer: \[ 1512 \]