Find the coefficient of x^7 in the expansion of (1-x) ¹²

Apply Binomial Theorem: We will use the binomial theorem to find the coefficient of x^7 in the expansion of (1-x)¹². The binomial theorem states that (a+b)ⁿ can be expanded as the sum of terms in the form of C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient ＂n choose k＂.Identify Term with k=7: To find the coefficient of x^7, we need to identify the term in the expansion where k=7. This term will be in the form of C(12, 7) * (1)^(12-7) * (-x)^7.Calculate Binomial Coefficient: Calculate the binomial coefficient C(12, 7). This is equal to 12! / (7! * (12-7)!).Perform Calculation: Perform the calculation: 12! / (7! * 5!) = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1) = 792.Simplify Term: Now, we have the term as 792 * (1)^5 * (-x)^7. Since (1)^5 is just 1, we can ignore it. The term simplifies to 792 * (-x)^7.Find Coefficient of x^7: The coefficient of x^7 is the number in front of the x^7 term, which is 792 multiplied by the sign of (-x)^7. Since 7 is odd, (-x)^7 is negative, so the coefficient is -792.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the coefficient of $x^7$ in the expansion of $(1-x)^{12}$

View step-by-step help

Home

Math Problems

Algebra 1

Evaluate integers raised to rational exponents

Full solution

Q. Find the coefficient of

x^7

in the expansion of

(1-x)^{12}

Apply Binomial Theorem: We will use the binomial theorem to find the coefficient of $x^7$ in the expansion of $(1-x)^{12}$ . The binomial theorem states that $(a+b)^n$ can be expanded as the sum of terms in the form of $C(n, k) \cdot a^{n-k} \cdot b^k$ , where $C(n, k)$ is the binomial coefficient ＂ $n$ choose $k$ ＂.
Identify Term with $k=7$ : To find the coefficient of $x^7$ , we need to identify the term in the expansion where $k=7$ . This term will be in the form of $C(12, 7) \cdot (1)^{(12-7)} \cdot (-x)^7$ .
Calculate Binomial Coefficient: Calculate the binomial coefficient $C(12, 7)$ . This is equal to $\frac{12!}{7! \cdot (12-7)!}$ .
Perform Calculation: Perform the calculation: $12! / (7! * 5!) = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1) = 792$ .
Simplify Term: Now, we have the term as $792 \times (1)^5 \times (-x)^7$ . Since $(1)^5$ is just $1$ , we can ignore it. The term simplifies to $792 \times (-x)^7$ .
Find Coefficient of $x^7$ : The coefficient of $x^7$ is the number in front of the $x^7$ term, which is $792$ multiplied by the sign of $(-x)^7$ . Since $7$ is odd, $(-x)^7$ is negative, so the coefficient is $-792$ .