The coefficient of x⁷ in the expansion of (1-x-x²+x³)⁶ is:

To find the coefficient of \( x^7 \) in the expansion of \( (1 - x - x^2 + x^3)^6 \), we can use the multinomial theorem. We need to expand \( (1 - x - x^2 + x^3)^6 \) and identify the terms that contribute to \( x^7 \).

### Step 1: Express the expansion in terms of the multinomial theorem.
The multinomial theorem for expanding \( (a + b + c + d)^n \) gives the general term:

\[
\text{Term} = \frac{6!}{k_1!k_2!k_3!k_4!} \cdot a^{k_1} b^{k_2} c^{k_3} d^{k_4}
\]
where \( k_1 + k_2 + k_3 + k_4 = 6 \), and the exponents of each term are powers of \( 1 \), \( -x \), \( -x^2 \), and \( x^3 \).

In our case:
- \( a = 1 \),
- \( b = -x \),
- \( c = -x^2 \),
- \( d = x^3 \).

### Step 2: Find combinations that contribute to \( x^7 \).
Each of the terms \( 1 \), \( -x \), \( -x^2 \), and \( x^3 \) contributes to different powers of \( x \). We need to find the combinations of powers \( k_1 \), \( k_2 \), \( k_3 \), and \( k_4 \) such that:

\[
k_2 + 2k_3 + 3k_4 = 7 \quad \text{and} \quad k_1 + k_2 + k_3 + k_4 = 6.
\]

We'll now determine all possible sets of non-negative integers \( (k_1, k_2, k_3, k_4) \) that satisfy these conditions.

### Step 3: Solve the system of equations.
Let's find the possible values of \( k_2 \), \( k_3 \), and \( k_4 \) that satisfy the equation \( k_2 + 2k_3 + 3k_4 = 7 \) under the constraint that \( k_1 + k_2 + k_3 + k_4 = 6 \).

#### Possible combinations:
- \( k_2 = 1 \), \( k_3 = 2 \), \( k_4 = 1 \)
- \( k_2 = 4 \), \( k_3 = 1 \), \( k_4 = 1 \)
- \( k_2 = 7 \), \( k_

The coefficient of x⁷ in the expansion of (1 - x - x² + x³)⁶ is:

Discover how to find the coefficient of x⁷ in the expansion of (1 - x - x² + x³)⁶ using the multinomial theorem. This step-by-step solution includes identifying relevant combinations and solving using algebraic and combinatoric methods, suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation