Question

The coefficient of \(x^4\) in \((1 + x + x^3 + x^4)^{10}\) is

• A. 210
• B. 100
• C. 310
• D. 110

Hint:

For multinomial expansions, list all exponent combinations carefully.

Answer 1

The Correct Option is C

Concept: Use multinomial expansion.

Step 1: Possible combinations.
We need total power of \(x\) = 4. Possible cases: \[ x^4 \text{ from } (x^4)^1 \] \[ x^3 \cdot x \] \[ x \cdot x \cdot x \cdot x \]

Step 2: Count contributions.
Case 1: \[ \binom{10}{1} = 10 \] Case 2: \[ \binom{10}{1,1,8} = 90 \] Case 3: \[ \binom{10}{4} = 210 \]

Step 3: Total.
\[ 10 + 90 + 210 = 310 \]