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 the expression \( (1 - 2x + x^3)^{10} \), we can use the multinomial expansion. The multinomial theorem states that:

\((a_1 + a_2 + \ldots + a_m)^n = \sum \frac{n!}{k_1! k_2! \ldots k_m!} a_1^{k_1} a_2^{k_2} \ldots a_m^{k_m}\)

where \( k_1 + k_2 + \ldots + k_m = n \). For our problem, this becomes:

\( (1 - 2x + x^3)^{10} = \sum_{k_1 + k_2 + k_3 = 10} \frac{10!}{k_1! k_2! k_3!} 1^{k_1} (-2x)^{k_2} (x^3)^{k_3} \)

The specific term from the expansion giving \( x^7 \) will be when:

\(k_2 + 3k_3 = 7\)\(k_1 + k_2 + k_3 = 10\)

Solving these equations, we have:

1. From \( k_2 + 3k_3 = 7 \), we can deduce possible values for \( k_3 \).

• \( k_3 = 0 \): \( k_2 = 7 \)
• \( k_3 = 1 \): \( k_2 = 4 \)
• \( k_3 = 2 \): \( k_2 = 1 \)

2. From \( k_1 + k_2 + k_3 = 10 \), determine \( k_1 \).

• \( k_3 = 0 \), \( k_2 = 7 \): \( k_1 = 3 \)
• \( k_3 = 1 \), \( k_2 = 4 \): \( k_1 = 5 \)
• \( k_3 = 2 \), \( k_2 = 1 \): \( k_1 = 7 \)

Now compute the coefficient for each case:

• For \( (k_1, k_2, k_3) = (3, 7, 0) \):
• For \( (k_1, k_2, k_3) = (5, 4, 1) \):
• For \( (k_1, k_2, k_3) = (7, 1, 2) \):

Sum these contributions for the coefficient of \( x^7 \):

\( 4080 = -15360 + 3360 + (-90) = 4080 \)

Therefore, the coefficient of \( x^7 \) is 4080.