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.