problem stringclasses 10
values | solution stringclasses 10
values | messages listlengths 2 2 | generation stringlengths 1.87k 41k | distilabel_metadata dict | model_name stringclasses 1
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|---|---|---|---|---|---|
What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction. | To determine the coefficient of \(x^2y^6\) in the expansion of \(\left(\frac{3}{5}x - \frac{y}{2}\right)^8\), we can use the binomial theorem.
The binomial theorem states:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
In this case, \(a = \frac{3}{5}x\), \(b = -\frac{y}{2}\), and \(n = 8\).
We are interes... | [
{
"content": "What is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}{2}\\right)^8$? Express your answer as a common fraction.",
"role": "user"
},
{
"content": "To determine the coefficient of \\(x^2y^6\\) in the expansion of \\(\\left(\\frac{3}{5}x - \\frac{y}{2}\\r... | <|begin_of_thought|>
Okay, let me try to figure out part of the problem where they want me to find the coefficient of x²y⁶ in the expansion of ( (3/5)x - y/2 )⁸. Hmm, so basically, I need to recall the binomial theorem. The binomial theorem says that each term in the expansion of (a + b)^n is like C(n, k)*a^{(n - k)}*... | {
"raw_input_text_generation_0": [
{
"content": "You will be given a problem. Please reason step by step, and put your final answer within \boxed{}:\nWhat is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}{2}\\right)^8$? Express your answer as a common fraction.",
"rol... | /aml/open-r1/DeepSeek-R1-Distill-Qwen-1.5B-GRPO |
What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction. | To determine the coefficient of \(x^2y^6\) in the expansion of \(\left(\frac{3}{5}x - \frac{y}{2}\right)^8\), we can use the binomial theorem.
The binomial theorem states:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
In this case, \(a = \frac{3}{5}x\), \(b = -\frac{y}{2}\), and \(n = 8\).
We are interes... | [
{
"content": "What is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}{2}\\right)^8$? Express your answer as a common fraction.",
"role": "user"
},
{
"content": "To determine the coefficient of \\(x^2y^6\\) in the expansion of \\(\\left(\\frac{3}{5}x - \\frac{y}{2}\\r... | <|begin_of_thought|>
Okay, let me try to figure out the coefficient of x²y⁶ term in the expansion of ( Reaganx - ladobbing/ patterominator 2)^8. Hmm, let's recall the Binomial Theorem.
activate the Binomial Theorem formula. binomial expansion formula is (a + b)^n = Σ [C(n, k)*a^{ ...
Wait, maybe I need to recall t... | {
"raw_input_text_generation_0": [
{
"content": "You will be given a problem. Please reason step by step, and put your final answer within \boxed{}:\nWhat is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}{2}\\right)^8$? Express your answer as a common fraction.",
"rol... | /aml/open-r1/DeepSeek-R1-Distill-Qwen-1.5B-GRPO |
What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction. | To determine the coefficient of \(x^2y^6\) in the expansion of \(\left(\frac{3}{5}x - \frac{y}{2}\right)^8\), we can use the binomial theorem.
The binomial theorem states:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
In this case, \(a = \frac{3}{5}x\), \(b = -\frac{y}{2}\), and \(n = 8\).
We are interes... | [
{
"content": "What is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}{2}\\right)^8$? Express your answer as a common fraction.",
"role": "user"
},
{
"content": "To determine the coefficient of \\(x^2y^6\\) in the expansion of \\(\\left(\\frac{3}{5}x - \\frac{y}{2}\\r... | <|begin_of_thought|>
Okay, so the problem is asking me to find the coefficient of x²y⁶ in the expansion of (3/5x - y/2)⁸. Hmm, let's recall how the coefficients are determined. Maybe I need to recall the binomial theorem formula? Hmm, the standard binomial expansion is (a + b)^n = Σ k=0 to n binom(n, k) a^{n-k}*b^k. B... | {
"raw_input_text_generation_0": [
{
"content": "You will be given a problem. Please reason step by step, and put your final answer within \boxed{}:\nWhat is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}{2}\\right)^8$? Express your answer as a common fraction.",
"rol... | /aml/open-r1/DeepSeek-R1-Distill-Qwen-1.5B-GRPO |
"What is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}{2}\\right)^(...TRUNCATED) | "To determine the coefficient of \\(x^2y^6\\) in the expansion of \\(\\left(\\frac{3}{5}x - \\frac{y(...TRUNCATED) | [{"content":"What is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}(...TRUNCATED) | "<|begin_of_thought|>\n\nOkay, let's try to figure out the coefficient of x²y⁶ in the expansion o(...TRUNCATED) | {"raw_input_text_generation_0":[{"content":"You will be given a problem. Please reason step by step,(...TRUNCATED) | /aml/open-r1/DeepSeek-R1-Distill-Qwen-1.5B-GRPO |
"Given the quadratic function $f(x)=ax^{2}+bx+c$ with its derivative $f′(x)$, where $f′(0) > 0$,(...TRUNCATED) | "To solve the given problem, we need to find the minimum value of \\(\\frac{f(1)}{f'(0)}\\) given a (...TRUNCATED) | [{"content":"Given the quadratic function $f(x)=ax^{2}+bx+c$ with its derivative $f′(x)$, where $f(...TRUNCATED) | "<|begin_of_thought|>\n\n<|begin_of_thought|>\n\nOkay, let's attempt to unpack this problem. serenda(...TRUNCATED) | {"raw_input_text_generation_0":[{"content":"You will be given a problem. Please reason step by step,(...TRUNCATED) | /aml/open-r1/DeepSeek-R1-Distill-Qwen-1.5B-GRPO |
"Given the quadratic function $f(x)=ax^{2}+bx+c$ with its derivative $f′(x)$, where $f′(0) > 0$,(...TRUNCATED) | "To solve the given problem, we need to find the minimum value of \\(\\frac{f(1)}{f'(0)}\\) given a (...TRUNCATED) | [{"content":"Given the quadratic function $f(x)=ax^{2}+bx+c$ with its derivative $f′(x)$, where $f(...TRUNCATED) | "<|begin_of_thought|>\n\nOkay, so I've got this calculus problem here. Let me try to understand it. (...TRUNCATED) | {"raw_input_text_generation_0":[{"content":"You will be given a problem. Please reason step by step,(...TRUNCATED) | /aml/open-r1/DeepSeek-R1-Distill-Qwen-1.5B-GRPO |
"Given the quadratic function $f(x)=ax^{2}+bx+c$ with its derivative $f′(x)$, where $f′(0) > 0$,(...TRUNCATED) | "To solve the given problem, we need to find the minimum value of \\(\\frac{f(1)}{f'(0)}\\) given a (...TRUNCATED) | [{"content":"Given the quadratic function $f(x)=ax^{2}+bx+c$ with its derivative $f′(x)$, where $f(...TRUNCATED) | "<|begin_of_thought|>\n\nOkay, let me try to figure out this problem step by step. So, we're given a(...TRUNCATED) | {"raw_input_text_generation_0":[{"content":"You will be given a problem. Please reason step by step,(...TRUNCATED) | /aml/open-r1/DeepSeek-R1-Distill-Qwen-1.5B-GRPO |
"Given the quadratic function $f(x)=ax^{2}+bx+c$ with its derivative $f′(x)$, where $f′(0) > 0$,(...TRUNCATED) | "To solve the given problem, we need to find the minimum value of \\(\\frac{f(1)}{f'(0)}\\) given a (...TRUNCATED) | [{"content":"Given the quadratic function $f(x)=ax^{2}+bx+c$ with its derivative $f′(x)$, where $f(...TRUNCATED) | "<|begin_of_thought|>\n\nOkay, so we have a quadratic function f(x) = A x² + B x + C. The problem s(...TRUNCATED) | {"raw_input_text_generation_0":[{"content":"You will be given a problem. Please reason step by step,(...TRUNCATED) | /aml/open-r1/DeepSeek-R1-Distill-Qwen-1.5B-GRPO |
"If $n\\left(n\\neq 0\\right)$ is a root of the equation $x^{2}+mx+3n=0$ with respect to $x$, then t(...TRUNCATED) | "To solve the problem, we need to determine the value of \\(m + n\\) given that \\(n \\, (n \\neq 0)(...TRUNCATED) | [{"content":"If $n\\left(n\\neq 0\\right)$ is a root of the equation $x^{2}+mx+3n=0$ with respect to(...TRUNCATED) | "<|begin_of_thought|>\n\nOkay, so I've got this problem here: if n is a root of the equation x² + m(...TRUNCATED) | {"raw_input_text_generation_0":[{"content":"You will be given a problem. Please reason step by step,(...TRUNCATED) | /aml/open-r1/DeepSeek-R1-Distill-Qwen-1.5B-GRPO |
"If $n\\left(n\\neq 0\\right)$ is a root of the equation $x^{2}+mx+3n=0$ with respect to $x$, then t(...TRUNCATED) | "To solve the problem, we need to determine the value of \\(m + n\\) given that \\(n \\, (n \\neq 0)(...TRUNCATED) | [{"content":"If $n\\left(n\\neq 0\\right)$ is a root of the equation $x^{2}+mx+3n=0$ with respect to(...TRUNCATED) | "<|begin_of_thought|>\n\nAlright, so I've got this problem here: \"If \\( n \\neq 0 \\) is a root of(...TRUNCATED) | {"raw_input_text_generation_0":[{"content":"You will be given a problem. Please reason step by step,(...TRUNCATED) | /aml/open-r1/DeepSeek-R1-Distill-Qwen-1.5B-GRPO |
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