| { | |
| "log_file": [ | |
| "step_124-b_0.log", | |
| "step_124-b_1.log", | |
| "step_124-b_2.log", | |
| "step_124-b_3.log", | |
| "step_124-b_4.log", | |
| "step_124-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:\nIn how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)\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:\nIn how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)\n\n### Response:\nSEARCH[\"\\\\cos(8)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/6190.json\n\n#### Similar Question\n[asy] draw((0,0)--(1,sqrt(3)),black+linewidth(.75),EndArrow); draw((0,0)--(1,-sqrt(3)),black+linewidth(.75),EndArrow); draw((0,0)--(1,0),dashed+black+linewidth(.75)); dot((1,0)); MP(\"P\",(1,0),E); [/asy]\nLet $S$ be the set of points on the rays forming the sides of a $120^{\\circ}$ angle, and let $P$ be a fixed point inside the angle on the angle bisector. Consider all distinct equilateral triangles $PQR$ with $Q$ and $R$ in $S$. (Points $Q$ and $R$ may be on the same ray, and switching the names of $Q$ and $R$ does not create a distinct triangle.) There are\n[asy] draw(circle((0,0),10),black+linewidth(.75)); draw((-10,0)--(10,0),black+linewidth(.75)); draw((-10,0)--(9,sqrt(19)),black+linewidth(.75)); draw((-10,0)--(9,-sqrt(19)),black+linewidth(.75)); draw((2,0)--(9,sqrt(19)),black+linewidth(.75)); draw((2,0)--(9,-sqrt(19)),black+linewidth(.75)); MP(\"X\",(2,0),N);MP(\"A\",(-10,0),W);MP(\"D\",(10,0),E);MP(\"B\",(9,sqrt(19)),E);MP(\"C\",(9,-sqrt(19)),E); [/asy]\nPoints $A,B,C$ and $D$ are on a circle of diameter $1$, and $X$ is on diameter $\\overline{AD}.$\nIf $BX=CX$ and $3\\angle{BAC}=\\angle{BXC}=36^\\circ$, then $AX=$\n$\\text{(A) } \\cos(6^\\circ)\\cos(12^\\circ)\\sec(18^\\circ)\\quad\\\\ \\text{(B) } \\cos(6^\\circ)\\sin(12^\\circ)\\csc(18^\\circ)\\quad\\\\ \\text{(C) } \\cos(6^\\circ)\\sin(12^\\circ)\\sec(18^\\circ)\\quad\\\\ \\text{(D) } \\sin(6^\\circ)\\sin(12^\\circ)\\csc(18^\\circ)\\quad\\\\ \\text{(E) } \\sin(6^\\circ)\\sin(12^\\circ)\\sec(18^\\circ)$\n\n#### User Answer\nWe have all the angles we need, but most obviously, we see that right angle in triangle $ABD$.\nNote also that angle $BAD$ is 6 degrees, so length $AB = cos(6)$ because the diameter, $AD$, is 1.\nNow, we can concentrate on triangle $ABX$ (after all, now we can decipher all angles easily and use Law of Sines).\nWe get:\n$\\frac{AB}{\\sin(\\angle{AXB})} =\\frac{AX}{\\sin(\\angle{ABX})}$\nThat's equal to\n$\\frac{\\cos(6)}{\\sin(180-18)} =\\frac{AX}{\\sin(12)}$\nTherefore, our answer is equal to: $\\cos(6^\\circ)\\sin(12^\\circ)\\csc(18^\\circ)$\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:\nIn how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)\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:\nIn how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)\n\n### Response:\nSEARCH[\"\\\\cos(8)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: /tuna1/scratch/w32zhong/corpus/MATH/train/geometry/6119.json\n\n#### Similar Question\nEquilateral triangle $T$ is inscribed in circle $A$, which has radius $10$. Circle $B$ with radius $3$ is internally tangent to circle $A$ at one vertex of $T$. Circles $C$ and $D$, both with radius $2$, are internally tangent to circle $A$ at the other two vertices of $T$. Circles $B$, $C$, and $D$ are all externally tangent to circle $E$, which has radius $\\dfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=4; pair A=(0,0), D=8*dir(330), C=8*dir(210), B=7*dir(90); pair Ep=(0,4-27/5); pair[] dotted={A,B,C,D,Ep}; draw(Circle(A,10)); draw(Circle(B,3)); draw(Circle(C,2)); draw(Circle(D,2)); draw(Circle(Ep,27/5)); dot(dotted); label(\"$E$\",Ep,E); label(\"$A$\",A,W); label(\"$B$\",B,W); label(\"$C$\",C,W); label(\"$D$\",D,E); [/asy]\n\n#### User Answer\nLet $X$ be the intersection of the circles with centers $B$ and $E$, and $Y$ be the intersection of the circles with centers $C$ and $E$. Since the radius of $B$ is $3$, $AX =4$. Assume $AE$ = $p$. Then $EX$ and $EY$ are radii of circle $E$ and have length $4+p$. $AC = 8$, and angle $CAE = 60$ degrees because we are given that triangle $T$ is equilateral. Using the Law of Cosines on triangle $CAE$, we obtain\n$(6+p)^2 =p^2 + 64 - 2(8)(p) \\cos 60$.\nThe $2$ and the $\\cos 60$ terms cancel out:\n$p^2 + 12p +36 = p^2 + 64 - 8p$\n$12p+ 36 = 64 - 8p$\n$p =\\frac {28}{20} = \\frac {7}{5}$. The radius of circle $E$ is $4 + \\frac {7}{5} = \\frac {27}{5}$, so the answer is $27 + 5 = 32$.\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:\nIn how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)\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:\nIn how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)\n\n### Response:\nSEARCH[\"\\\\cos(8)\"]\n\nHere are the results:\n--- RESULTS BEGIN ---\nURL: Rectangle $ABCD$ and a semicircle with diameter $AB$ are coplanar and have nonoverlapping interiors. Let $\\mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $\\ell$ meets the semicircle, segment $AB$, and segment $CD$ at distinct points $N$, $U$, and $T$, respectively. Line $\\ell$ divides region $\\mathcal{R}$ into two regions with areas in the ratio $1: 2$. Suppose that $AU = 84$, $AN = 126$, and $UB = 168$. Then $DA$ can be represented as $m\\sqrt {n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.\n[asy] import graph; defaultpen(linewidth(0.7)+fontsize(10)); size(500); pen zzttqq = rgb(0.6,0.2,0); pen xdxdff = rgb(0.4902,0.4902,1); /* segments and figures */ draw((0,-154.31785)--(0,0)); draw((0,0)--(252,0)); draw((0,0)--(126,0),zzttqq); draw((126,0)--(63,109.1192),zzttqq); draw((63,109.1192)--(0,0),zzttqq); draw((-71.4052,(+9166.01287-109.1192*-71.4052)/21)--(504.60925,(+9166.01287-109.1192*504.60925)/21)); draw((0,-154.31785)--(252,-154.31785)); draw((252,-154.31785)--(252,0)); draw((0,0)--(84,0)); draw((84,0)--(252,0)); draw((63,109.1192)--(63,0)); draw((84,0)--(84,-154.31785)); draw(arc((126,0),126,0,180)); /* points and labels */ dot((0,0)); label(\"$A$\",(-16.43287,-9.3374),NE/2); dot((252,0)); label(\"$B$\",(255.242,5.00321),NE/2); dot((0,-154.31785)); label(\"$D$\",(3.48464,-149.55669),NE/2); dot((252,-154.31785)); label(\"$C$\",(255.242,-149.55669),NE/2); dot((126,0)); label(\"$O$\",(129.36332,5.00321),NE/2); dot((63,109.1192)); label(\"$N$\",(44.91307,108.57427),NE/2); label(\"$126$\",(28.18236,40.85473),NE/2); dot((84,0)); label(\"$U$\",(87.13819,5.00321),NE/2); dot((113.69848,-154.31785)); label(\"$T$\",(116.61611,-149.55669),NE/2); dot((63,0)); label(\"$N'$\",(66.42398,5.00321),NE/2); label(\"$84$\",(41.72627,-12.5242),NE/2); label(\"$168$\",(167.60494,-12.5242),NE/2); dot((84,-154.31785)); label(\"$T'$\",(87.13819,-149.55669),NE/2); dot((252,0)); label(\"$I$\",(255.242,5.00321),NE/2); clip((-71.4052,-225.24323)--(-71.4052,171.51361...\n\nthe center of the semicircl is also the midpoint of $AB$. let thi point be O . let $h$ be the length of $AD$. rescal everyth by 42, so $AU = 2, AN = 3, UB = 4$. then $AB = 6$ so $OA = OB = 3$. sinc $ON$ is a radiu of the semicircl , $ON = 3$. thu $OAN$ is an equilater triangl . let $X$, $Y$, and $Z$ be the area of triangl $OUN$, sector $ONB$, and trapezoid $UBCT$ respect.<em class=\"hl\">$X = \\frac {1}{2}(UO)(NO)\\" | |
| ], | |
| "response": [ | |
| " SEARCH[\"\\\\cos(8)\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThere are 8 people to choose from for the president, and then 7 people to choose from for the 2-person committee. Thus, there are $8 \\times 7 = \\boxed{56}$ ways to choose the president and the 2-person committee.</s>", | |
| " SEARCH[\"\\\\cos(8)\"]</s>", | |
| " The result looks irrelevant, I will completely ignore it and answer the question directly.\n\nThere are 8 people to choose from for president, and then 7 people to choose from for the 2-person committee. Thus, there are $8 \\times 7 = \\boxed{56}$ ways to choose the president and the committee.</s>", | |
| " SEARCH[\"\\\\cos(8)\"]</s>", | |
| "cos" | |
| ], | |
| "src_path": [ | |
| "train/prealgebra/236.json", | |
| "train/prealgebra/236.json", | |
| "train/prealgebra/236.json", | |
| "train/prealgebra/236.json", | |
| "train/prealgebra/236.json", | |
| "train/prealgebra/236.json" | |
| ], | |
| "input": [ | |
| "In how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)", | |
| "In how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)", | |
| "In how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)", | |
| "In how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)", | |
| "In how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)", | |
| "In how many ways can a president and a 2-person committee be chosen from a group of 8 people (where the order in which we choose the 2 people doesn't matter)? (The president cannot be on the committee.)" | |
| ], | |
| "output": [ | |
| "We have to first choose a president, and then we have to choose 2 people, but the order in which we choose the people doesn't matter. So first there are 8 ways to choose the president. Then there are 7 ways to choose the first person, and 6 ways to choose second person. However, we have overcounted, since choosing person A first and person B second will give us the same committee as choosing person B first and person A second. Each committee is counted twice in our original $7 \\times 6$ count, so we must divide by 2 to correct for this overcount, giving us $8\\times(7 \\times 6)/2 = \\boxed{168}$ ways to choose a president and a 2-person committee from 8 people.", | |
| "We have to first choose a president, and then we have to choose 2 people, but the order in which we choose the people doesn't matter. So first there are 8 ways to choose the president. Then there are 7 ways to choose the first person, and 6 ways to choose second person. However, we have overcounted, since choosing person A first and person B second will give us the same committee as choosing person B first and person A second. Each committee is counted twice in our original $7 \\times 6$ count, so we must divide by 2 to correct for this overcount, giving us $8\\times(7 \\times 6)/2 = \\boxed{168}$ ways to choose a president and a 2-person committee from 8 people.", | |
| "We have to first choose a president, and then we have to choose 2 people, but the order in which we choose the people doesn't matter. So first there are 8 ways to choose the president. Then there are 7 ways to choose the first person, and 6 ways to choose second person. However, we have overcounted, since choosing person A first and person B second will give us the same committee as choosing person B first and person A second. Each committee is counted twice in our original $7 \\times 6$ count, so we must divide by 2 to correct for this overcount, giving us $8\\times(7 \\times 6)/2 = \\boxed{168}$ ways to choose a president and a 2-person committee from 8 people.", | |
| "We have to first choose a president, and then we have to choose 2 people, but the order in which we choose the people doesn't matter. So first there are 8 ways to choose the president. Then there are 7 ways to choose the first person, and 6 ways to choose second person. However, we have overcounted, since choosing person A first and person B second will give us the same committee as choosing person B first and person A second. Each committee is counted twice in our original $7 \\times 6$ count, so we must divide by 2 to correct for this overcount, giving us $8\\times(7 \\times 6)/2 = \\boxed{168}$ ways to choose a president and a 2-person committee from 8 people.", | |
| "We have to first choose a president, and then we have to choose 2 people, but the order in which we choose the people doesn't matter. So first there are 8 ways to choose the president. Then there are 7 ways to choose the first person, and 6 ways to choose second person. However, we have overcounted, since choosing person A first and person B second will give us the same committee as choosing person B first and person A second. Each committee is counted twice in our original $7 \\times 6$ count, so we must divide by 2 to correct for this overcount, giving us $8\\times(7 \\times 6)/2 = \\boxed{168}$ ways to choose a president and a 2-person committee from 8 people.", | |
| "We have to first choose a president, and then we have to choose 2 people, but the order in which we choose the people doesn't matter. So first there are 8 ways to choose the president. Then there are 7 ways to choose the first person, and 6 ways to choose second person. However, we have overcounted, since choosing person A first and person B second will give us the same committee as choosing person B first and person A second. Each committee is counted twice in our original $7 \\times 6$ count, so we must divide by 2 to correct for this overcount, giving us $8\\times(7 \\times 6)/2 = \\boxed{168}$ ways to choose a president and a 2-person committee from 8 people." | |
| ], | |
| "rewards": [ | |
| 0.0, | |
| 0.0, | |
| 0.0, | |
| 0.0, | |
| 0.0, | |
| 0.0 | |
| ] | |
| } |