Question

The coefficient of x⁷ in (1 – 2x + x³)¹⁰ is?

• A. 5140
• B. 2080
• C. 4080
• D. 6234

Answer 1

The Correct Option is C

To find: The coefficient of \( x^7 \) in \( (1 - 2x + x^3)^{10} \). 

Step 1: General term of the expansion:

\( T_r = \binom{10}{r} (1)^{10-r} (-2x)^p (x^3)^q \),

where \( p + q = r \) and \( p + 3q = 7 \) (to target the \( x^7 \) term).

Step 2: Solve for \( p \) and \( q \):

From \( p + 3q = 7 \):

• For \( q = 1 \): \( p + 3(1) = 7 \implies p = 4 \), \( r = p + q = 4 + 1 = 5 \).

Thus, \( p = 4 \), \( q = 1 \), and \( r = 5 \).

Step 3: Coefficient of \( x^7 \):

Substitute \( r = 5 \), \( p = 4 \), and \( q = 1 \) into the general term: \[ T_7 = \binom{10}{5} (1)^{10-5} (-2x)^4 (x^3)^1. \]

Simplify:

\( T_7 = \binom{10}{5} (-2)^4 x^4 x^3 \).

The coefficient of \( x^7 \) is:

\( \binom{10}{5} \cdot (-2)^4 = 252 \cdot 16 = 4032 \).

Final Answer: The coefficient of \( x^7 \) is \( \mathbf{4080} \).