| { | |
| "log_file": [ | |
| "step_132-b_0.log", | |
| "step_132-b_1.log", | |
| "step_132-b_2.log", | |
| "step_132-b_3.log", | |
| "step_132-b_4.log", | |
| "step_132-b_5.log" | |
| ], | |
| "query": [ | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\n2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?\n\n### Response:\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\n2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?\n\n### Response:\nSEARCH[\"2 diagonals of a regular 9-sided polygon are chosen.\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/counting_and_probability/902.json\n\n#### Similar Question\n2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?\n\n#### User Answer\nThere are $\\binom{9}{2} = 36$ pairs of points in the nonagon, and all but 9 (the sides of the nonagon) are diagonals, which means there are 27 diagonals. So there are $\\binom{27}{2} = 351$ pairs of diagonals. Any four points on the nonagon uniquely determine a pair of intersecting diagonals. (If vertices $A,B,C,D$ are chosen, where $ABCD$ is a convex quadrilateral, the intersecting pair of diagonals are $AC$ and $BD$.) So the number of sets of intersecting diagonals is the number of combinations of 4 points, which is $\\binom{9}{4} = 126$. So the probability that a randomly chosen pair of diagonals intersect is $\\dfrac{126}{351} = \\dfrac{14}{39}$.\n\n--- RESULTS END ---\n\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\n2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?\n\n### Response:\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\n2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?\n\n### Response:\nSEARCH[\"2 diagonals of a regular 9-sided polygon are chosen.\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/algebra/556.json\n\n#### Similar Question\nIf a regular polygon has a total of nine diagonals, how many sides does it have?\n\n#### User Answer\nGiven the number of sides of a polygon, $n$, the number of diagonals is given by $D=\\frac{n(n-3)}{2}$. To find the number of sides given the number of diagonals, we can solve this equation for $n$. \\begin{align*}D&=\\frac{n(n-3)}{2} \\\\ 2D&=n^2-3n \\\\ 0&=n^2-3n-2D.\\end{align*} Then, using the quadratic formula we have $n=\\frac{3\\pm\\sqrt{3^2-4(1)(-2D)}}{2(1)}=\\frac{3\\pm\\sqrt{9+8D}}{2}$.\n\nSince we are given that $D=9$, we have $n=\\frac{3\\pm\\sqrt{9+8(9)}}{2}=\\frac{3\\pm9}{2}=-3\\text{ or }6$. Since we must have a positive number of sides, a polygon with nine diagonals has $6$ sides.\n\n--- RESULTS END ---\n\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\n2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?\n\n### Response:\n", | |
| "<s>Below is an Instruction section that describes a task, paired with an Input section that provides further context.\nWrite in the Response section that appropriately completes the request.\n\n### Instruction:\nAnswer a math question in the input.\n\nTo assist you, you can invoke a math-aware search API (i.e., SEARCH) or a computation API (COMPUTE), and I will insert the returned API results for you right after each valid SEARCH or COMPUTE calls.\n\nThe SEARCH API is followed by its parameters which are a list of keywords in JSON format, for example:\n\nSEARCH[\"$x^2 = -1$\", \"imaginary numbers\"]\n\nDO NOT mix text and math in one JSON item, i.e. instead of writing:\n\nSEARCH['$what kind of curve is defined by x^2 - y^2 = 4$']\n\nwrite keyword by keyword with only one type in each:\n\nSEARCH[\"curve\", \"defined by\", \"$x^2 - y^2 = 4$\"]\n\nFor the COMPUTE API, it is also followed by its parameters in JSON. The first parameter `mode' is chosen from `calculate', `simplify' or `solve *', whereas the second parameter is the symbolic expression in LaTeX.\n\nFor example, to calculate sine of 270 degree, you can do:\n\nCOMPUTE[\"calculate\", \"\\\\sin(270 \\\\times \\\\frac{\\\\pi}{180})\"]\n\nTo simplify $\\sin^2 x + \\cos^2 x$, you can do:\n\nCOMPUTE[\"simplify\", \"\\\\sin^2(x) + \\\\cos^2(x)\"]\n\nAnd to solve $y = 1 - 2 y^2$ for y, you can do:\n\nCOMPUTE[\"solve y\", \"y = 1 - 2 y^2\"]\n\nFor the SEARCH API, only consider helpful API results for your goal, ignore irrelevant ones.\nFor the COMPUTE API, remember it is limited to simple tasks. It does not support linear algebra, nor matrix manipulations.\n\nWhen the API result is helpful, you can just rely on the result or extract the final answer from it directly, in such case, there is no need to answer from the begining and redo any existing derivations in the result.\n\nWhen API results are not helpful, ignore the results and answer the given math question directly!\n\nAt the end, indicate your final answer in boxed LaTeX. For example, if you think the final answer is \\sqrt{3}, write it as \\boxed{\\sqrt{3}} (in boxed LaTeX) at the very end of your output.\n\nTake a deep breath and now I will hand the math question to you!\n\n### Input:\n2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?\n\n### Response:\nSEARCH[\"2 diagonals of a regular 9-sided polygon are chosen.\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/1145.json\n\n#### Similar Question\nA regular polygon has an exterior angle that measures $15$ degrees. How many sides does the polygon have?\n\n#### User Answer\nThe sum of the exterior angles of a polygon is $360^\\circ$ as long as we take only one exterior angle per vertex. The polygon is regular, so all of the exterior angles have the same degree measure of $15$ degrees. If the polygon has $n$ sides, then the sum of the exterior angles is $15n=360$. So $n=24$ and the polygon has $24$ sides.\n\n--- RESULTS END ---\n\n" | |
| ], | |
| "response": [ | |
| " SEARCH[\"2 diagonals of a regular 9-sided polygon are chosen.\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe consider the two diagonals to be line segments.\n\nIf we draw the two diagonals, we obtain a rectangle that encloses a total of $9$ points.\n\n[asy]\nunitsize(0.3 cm);\n\nint i;\n\ndraw((0,0)--(9,0)--(9,8)--(0,8)--cycle);\ndraw((0,0)--(4,4)--(8,4)--(4,0)--cycle,dashed);\ndraw((4,4)--(5,5)--(8,5)--(5,4)--cycle,dashed);\n\nfor (i = 0; i <= 8; ++i) {\n\ndot((i,i));\n}\n\nlabel(\"$A$\", (0,0), S);\nlabel(\"$B$\", (9,0), S);\nlabel(\"$C$\", (9,8), NW);\nlabel(\"$D$\", (0,8), SE);\n[/asy]\n\nThe desired event happens if and only if the rectangle does not enclose any other points of the nonagon.\n\nWe can choose $2$ of the $9$ points that do not lie on a side of the rectangle in $\\binom{9}{2} = 36$ ways.\n\nWe can choose $2$ of the $8$ points that lie on a side of the rectangle in $\\binom{8}{2} = 28$ ways.\n\nThus, the desired event happens in $36 - 28 = 8$ ways.\n\nTherefore, the probability that the intersection of the two diagonals lies inside the nonagon is $\\boxed{\\dfrac{8}{36}}$.</s>", | |
| " SEARCH[\"2 diagonals of a regular 9-sided polygon are chosen.\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThe diagonals intersect in $\\frac{9-3}{2}=3$ triangles that do not share an edge. Each of these triangles has an area of $\\frac{1}{2}(9)(6)/2=9$. The total area of the triangles is $3(9)=27$. The total area of the nonagon is $9(9)=81$. The probability that the intersection lies inside the nonagon is $\\frac{27}{81}=\\boxed{\\frac{9}{27}}$.</s>", | |
| " SEARCH[\"2 diagonals of a regular 9-sided polygon are chosen.\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nWe first consider the special case where the two diagonals intersect at a vertex of the nonagon. Then the intersection lies inside the nonagon if and only if the vertex is part of a convex $2$-gon. There are $\\dbinom{9}{2} = 36$ ways to choose the vertex, and there are 6 vertices which are part of a convex $2$-gon. Thus the probability is $\\dfrac{6}{36} = \\dfrac{1}{6}$.\n\nNext we consider the case where the two diagonals intersect at a point which is not a vertex of the nonagon. Then the intersection lies inside the nonagon if and only if the point is part of a convex $2$-gon which is entirely contained within the nonagon. There are $\\dbinom{6}{2} = 15$ ways to choose such a $2$-gon. Since the nonagon has 9 sides, there are $9 - 2 = 7$ sides which are not part of the chosen $2$-gon, so there are $7 \\times 6 = 42$ ways to choose the points which are not in the $2$-gon. Thus the probability is $\\dfrac{15}{42} = \\boxed{\\dfrac{5}{14}}$.</s>" | |
| ], | |
| "src_path": [ | |
| "train/counting_and_probability/902.json", | |
| "train/counting_and_probability/902.json", | |
| "train/counting_and_probability/902.json", | |
| "train/counting_and_probability/902.json", | |
| "train/counting_and_probability/902.json", | |
| "train/counting_and_probability/902.json" | |
| ], | |
| "input": [ | |
| "2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?", | |
| "2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?", | |
| "2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?", | |
| "2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?", | |
| "2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?", | |
| "2 diagonals of a regular nonagon (a 9-sided polygon) are chosen. What is the probability that their intersection lies inside the nonagon?" | |
| ], | |
| "output": [ | |
| "There are $\\binom{9}{2} = 36$ pairs of points in the nonagon, and all but 9 (the sides of the nonagon) are diagonals, which means there are 27 diagonals. So there are $\\binom{27}{2} = 351$ pairs of diagonals. Any four points on the nonagon uniquely determine a pair of intersecting diagonals. (If vertices $A,B,C,D$ are chosen, where $ABCD$ is a convex quadrilateral, the intersecting pair of diagonals are $AC$ and $BD$.) So the number of sets of intersecting diagonals is the number of combinations of 4 points, which is $\\binom{9}{4} = 126$. So the probability that a randomly chosen pair of diagonals intersect is $\\dfrac{126}{351} = \\boxed{\\dfrac{14}{39}}$.", | |
| "There are $\\binom{9}{2} = 36$ pairs of points in the nonagon, and all but 9 (the sides of the nonagon) are diagonals, which means there are 27 diagonals. So there are $\\binom{27}{2} = 351$ pairs of diagonals. Any four points on the nonagon uniquely determine a pair of intersecting diagonals. (If vertices $A,B,C,D$ are chosen, where $ABCD$ is a convex quadrilateral, the intersecting pair of diagonals are $AC$ and $BD$.) So the number of sets of intersecting diagonals is the number of combinations of 4 points, which is $\\binom{9}{4} = 126$. So the probability that a randomly chosen pair of diagonals intersect is $\\dfrac{126}{351} = \\boxed{\\dfrac{14}{39}}$.", | |
| "There are $\\binom{9}{2} = 36$ pairs of points in the nonagon, and all but 9 (the sides of the nonagon) are diagonals, which means there are 27 diagonals. So there are $\\binom{27}{2} = 351$ pairs of diagonals. Any four points on the nonagon uniquely determine a pair of intersecting diagonals. (If vertices $A,B,C,D$ are chosen, where $ABCD$ is a convex quadrilateral, the intersecting pair of diagonals are $AC$ and $BD$.) So the number of sets of intersecting diagonals is the number of combinations of 4 points, which is $\\binom{9}{4} = 126$. So the probability that a randomly chosen pair of diagonals intersect is $\\dfrac{126}{351} = \\boxed{\\dfrac{14}{39}}$.", | |
| "There are $\\binom{9}{2} = 36$ pairs of points in the nonagon, and all but 9 (the sides of the nonagon) are diagonals, which means there are 27 diagonals. So there are $\\binom{27}{2} = 351$ pairs of diagonals. Any four points on the nonagon uniquely determine a pair of intersecting diagonals. (If vertices $A,B,C,D$ are chosen, where $ABCD$ is a convex quadrilateral, the intersecting pair of diagonals are $AC$ and $BD$.) So the number of sets of intersecting diagonals is the number of combinations of 4 points, which is $\\binom{9}{4} = 126$. So the probability that a randomly chosen pair of diagonals intersect is $\\dfrac{126}{351} = \\boxed{\\dfrac{14}{39}}$.", | |
| "There are $\\binom{9}{2} = 36$ pairs of points in the nonagon, and all but 9 (the sides of the nonagon) are diagonals, which means there are 27 diagonals. So there are $\\binom{27}{2} = 351$ pairs of diagonals. Any four points on the nonagon uniquely determine a pair of intersecting diagonals. (If vertices $A,B,C,D$ are chosen, where $ABCD$ is a convex quadrilateral, the intersecting pair of diagonals are $AC$ and $BD$.) So the number of sets of intersecting diagonals is the number of combinations of 4 points, which is $\\binom{9}{4} = 126$. So the probability that a randomly chosen pair of diagonals intersect is $\\dfrac{126}{351} = \\boxed{\\dfrac{14}{39}}$.", | |
| "There are $\\binom{9}{2} = 36$ pairs of points in the nonagon, and all but 9 (the sides of the nonagon) are diagonals, which means there are 27 diagonals. So there are $\\binom{27}{2} = 351$ pairs of diagonals. Any four points on the nonagon uniquely determine a pair of intersecting diagonals. (If vertices $A,B,C,D$ are chosen, where $ABCD$ is a convex quadrilateral, the intersecting pair of diagonals are $AC$ and $BD$.) So the number of sets of intersecting diagonals is the number of combinations of 4 points, which is $\\binom{9}{4} = 126$. So the probability that a randomly chosen pair of diagonals intersect is $\\dfrac{126}{351} = \\boxed{\\dfrac{14}{39}}$." | |
| ], | |
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