what is the coefficient of : a. x ^ 15 in x ^ 2 * (1 - 2x) ^ 21

To find the coefficient of \( x^{15} \) in the expression \( x^2 \times (1 - 2x)^{21} \), we can use the binomial theorem.

### Step 1: Expand the Binomial
The binomial theorem states that:
\[
(1 - 2x)^{21} = \sum_{k=0}^{21} \binom{21}{k} (-2x)^k
\]
Expanding this, we get:
\[
(1 - 2x)^{21} = \sum_{k=0}^{21} \binom{21}{k} (-2)^k x^k
\]

### Step 2: Include \( x^2 \)
Now, multiply the binomial expansion by \( x^2 \):
\[
x^2 \times (1 - 2x)^{21} = x^2 \times \sum_{k=0}^{21} \binom{21}{k} (-2)^k x^k
\]
This simplifies to:
\[
\sum_{k=0}^{21} \binom{21}{k} (-2)^k x^{k+2}
\]

### Step 3: Identify the Power of \( x \)
We want the coefficient of \( x^{15} \). In the sum, the general term is \( \binom{21}{k} (-2)^k x^{k+2} \). We need to find \( k \) such that:
\[
k + 2 = 15
\]
Solving for \( k \):
\[
k = 13
\]

### Step 4: Find the Coefficient
The coefficient of \( x^{15} \) is given by the term when \( k = 13 \):
\[
\binom{21}{13} (-2)^{13}
\]
First, calculate \( \binom{21}{13} \), which is equal to \( \binom{21}{8} \) (since \( \binom{n}{k} = \binom{n}{n-k} \)).

Now, calculate the coefficient:
\[
\binom{21}{8} \times (-2)^{13}
\]

Let's compute the exact value.

### Step 5: Final Computation
I'll calculate it now.The coefficient of \( x^{15} \) in the expression \( x^2 \times (1 - 2x)^{21} \) is \( -1,666,990,080 \).

Do you want more details on this, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. What is the coefficient of \( x^{10} \) in \( x^2 \times (1 - 2x)^{21} \)?
2. How would you find the coefficient of \( x^{15} \) if the binomial was \( (1 + 2x)^{21} \) instead?
3. Can you determine the constant term in the expansion of \( x^2 \times (1 - 2x)^{21} \)?
4. How does the binomial theorem apply to negative powers of \( x \)?
5. What would the expansion look like if the exponent was different, say \( (1 - 2x)^{10} \)?

**Tip:** When solving problems involving binomials, it's often easier to find the relevant terms directly rather than expanding the entire series.

What is the coefficient of x^15 in x^2 * (1 - 2x)^21?

Explore how to find the coefficient of x^15 in the expression x^2 * (1 - 2x)^21 using the binomial theorem. This detailed solution walks through expanding the binomial and applying the correct formula. Suitable for advanced high school students in Grades 10-12.

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