{"id":31195,"date":"2021-03-28T13:00:06","date_gmt":"2021-03-28T13:00:06","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=31195"},"modified":"2025-01-12T13:10:30","modified_gmt":"2025-01-12T13:10:30","slug":"%e7%83%ad%e5%8a%9b%e5%ad%a6%e4%b8%ad%e7%9a%84%e7%bb%84%e5%90%88%e9%97%ae%e9%a2%98","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/zh\/%e7%83%ad%e5%8a%9b%e5%ad%a6%e4%b8%ad%e7%9a%84%e7%bb%84%e5%90%88%e9%97%ae%e9%a2%98\/","title":{"rendered":"\u70ed\u529b\u5b66\u4e2d\u7684\u7ec4\u5408\u95ee\u9898"},"content":{"rendered":"<style>\np, ul, ol {\ntext-align: justify;\n}\nh1, h2 {\ntext-align:center;\n}\n<\/style>\n<h1>\u70ed\u529b\u5b66\u4e2d\u7684\u7ec4\u5408\u95ee\u9898<\/h1>\n<p style=\"text-align:center;\"><em>\u4e00\u4e2a\u7531\u6570\u767e\u4e07\u4e2a\u5143\u7d20\u7ec4\u6210\u7684\u7269\u7406\u7cfb\u7edf\u6709\u591a\u5c11\u79cd\u4e0d\u540c\u7684\u7ec4\u7ec7\u65b9\u5f0f\uff1f\u5728\u672c\u8282\u8bfe\u4e2d\uff0c\u6211\u4eec\u5c06\u63a2\u8ba8\u6570\u5b66\u5982\u4f55\u5728\u70ed\u529b\u5b66\u80cc\u666f\u4e0b\u56de\u7b54\u6b64\u7c7b\u95ee\u9898\uff0c\u4ece\u539f\u5b50\u7cfb\u7edf\u4e2d\u80fd\u91cf\u91cf\u5b50\u7684\u5206\u5e03\u5230\u5927\u89c4\u6a21\u7cfb\u7edf\u7684\u53ef\u80fd\u914d\u7f6e\u8ba1\u7b97\u3002\u6211\u4eec\u5c06\u5229\u7528\u7ec4\u5408\u6570\u5b66\u3001\u5bf9\u6570\u4ee5\u53ca\u65af\u7279\u6797\u516c\u5f0f\u7b49\u5de5\u5177\uff0c\u63a2\u7d22\u5982\u4f55\u5904\u7406\u6781\u5176\u5e9e\u5927\u7684\u6570\u5b57\u5e76\u89e3\u51b3\u770b\u4f3c\u96be\u4ee5\u653b\u514b\u7684\u95ee\u9898\u3002<\/em><\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong>\u5b8c\u6210\u672c\u8282\u8bfe\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a<\/p>\n<ol>\n<li><strong>\u7406\u89e3<\/strong>\u7ec4\u5408\u95ee\u9898\u5982\u4f55\u5728\u70ed\u529b\u5b66\u80cc\u666f\u4e0b\u5e94\u7528\uff0c\u5c24\u5176\u662f\u5728\u7269\u7406\u7cfb\u7edf\u7684\u7ec4\u7ec7\u4e2d\u3002<\/li>\n<li><strong>\u8ba1\u7b97<\/strong>\u901a\u8fc7\u7ec4\u5408\u6570\u65b9\u6cd5\u6c42\u89e3\u539f\u5b50\u7cfb\u7edf\u7684\u53ef\u80fd\u914d\u7f6e\u3002<\/li>\n<li><strong>\u5e94\u7528<\/strong>\u65af\u7279\u6797\u516c\u5f0f\u4f30\u7b97\u590d\u6742\u914d\u7f6e\u7684\u6570\u91cf\u7ea7\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><u>\u5185\u5bb9\u76ee\u5f55<\/u>:<br \/>\n<a href=\"#1\"><strong>\u7ec4\u5408\u95ee\u9898<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>\u6d89\u53ca\u5927\u6570\u7684\u95ee\u9898<\/strong><\/a><br \/>\n<a href=\"#3\">\u4f7f\u7528\u5bf9\u6570\u548c\u65af\u7279\u6797\u516c\u5f0f\u8fdb\u884c\u6570\u91cf\u7ea7\u8ba1\u7b97<\/a><br \/>\n<a href=\"#4\"><em>\u901a\u8fc7\u7b80\u5316\u8fd1\u4f3c\u6cd5\u5c55\u5f00<\/em><\/a><br \/>\n<a href=\"#5\"><em>\u901a\u8fc7\u5e38\u89c4\u8fd1\u4f3c\u6cd5\u5c55\u5f00<\/em><\/a><br \/>\n<a href=\"#6\"><strong>\u7ec4\u5408\u8ba1\u7b97\u4e0e\u6570\u91cf\u7ea7\u8ba1\u7b97\u793a\u4f8b<\/strong><\/a><br \/>\n<a href=\"#7\">\u6848\u4f8b 1: \u5927\u9636\u4e58<\/a><br \/>\n<a href=\"#8\">\u6848\u4f8b 2: \u5927\u7ec4\u5408<\/a>\n<\/p>\n<p>\u5728\u67d0\u4e9b\u7269\u7406\u60c5\u5883\u4e2d\uff0c\u4e00\u4e2a\u5e38\u89c1\u95ee\u9898\u662f\uff1a\u7ed9\u5b9a\u7684\u7cfb\u7edf\u53ef\u4ee5\u4ee5\u591a\u5c11\u79cd\u4e0d\u540c\u65b9\u5f0f\u8fdb\u884c\u7ec4\u7ec7\uff1f\u8fd9\u4e9b\u7ec4\u5408\u95ee\u9898\u5728\u70ed\u529b\u5b66\u4e2d\u975e\u5e38\u5e38\u89c1\u3002\u5c3d\u7ba1\u4e4d\u770b\u4e4b\u4e0b\u8fd9\u4e9b\u95ee\u9898\u5f88\u7b80\u5355\uff0c\u4f46\u4e00\u65e6\u6d89\u53ca\u5230\u5982<strong><a href=\"http:\/\/toposuranos.com\/material\/mol-y-masa-molar\/\" rel=\"noopener\" target=\"_blank\">\u963f\u4f0f\u4f3d\u5fb7\u7f57\u6570<\/a><\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">N_A<\/span> \u8fd9\u6837\u7684\u6781\u5927\u6570\u503c\u65f6\uff0c\u5904\u7406\u8fd9\u4e9b\u95ee\u9898\u7684\u590d\u6742\u6027\u5c31\u4f1a\u9aa4\u7136\u589e\u52a0\uff0c\u5c55\u73b0\u51fa\u89e3\u51b3\u6b64\u7c7b\u95ee\u9898\u7684\u6570\u5b66\u6311\u6218\u3002<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u7ec4\u5408\u95ee\u9898<\/h2>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/K7_De9wux4A?si=q93_T8xh15EdfJ6B\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><\/p>\n<p>\u4e3a\u4e86\u7406\u89e3\u6d89\u53ca\u70ed\u529b\u5b66\u4e2d\u7ec4\u5408\u95ee\u9898\u7684\u89c4\u6a21\uff0c\u6211\u4eec\u6765\u770b\u4e0b\u9762\u8fd9\u4e2a\u4f8b\u5b50\uff1a<\/p>\n<h4>\u793a\u4f8b\uff1a\u80fd\u91cf\u91cf\u5b50\u7684\u5206\u5e03\u7ec4\u5408<\/h4>\n<p>\u5047\u8bbe\u4e00\u4e2a\u7531 10 \u4e2a\u539f\u5b50\u7ec4\u6210\u7684\u7cfb\u7edf\u3002\u6bcf\u4e2a\u539f\u5b50\u53ea\u80fd\u5b58\u50a8 1 \u6216 0 \u4e2a\u80fd\u91cf\u5355\u4f4d\uff0c\u79f0\u4e3a<strong>\u80fd\u91cf\u91cf\u5b50<\/strong>\u3002\u5982\u679c\u6211\u4eec\u6709 (a) 10 \u4e2a\u80fd\u91cf\u91cf\u5b50\u548c (b) 5 \u4e2a\u80fd\u91cf\u91cf\u5b50\uff0c\u95ee\u8fd9\u4e9b\u91cf\u5b50\u53ef\u4ee5\u4ee5\u591a\u5c11\u79cd\u4e0d\u540c\u7684\u65b9\u5f0f\u5206\u5e03\uff1f<\/p>\n<h5>\u89e3\u7b54<\/h5>\n<p>\u6211\u4eec\u5c06\u539f\u5b50\u8868\u793a\u4e3a\u53ef\u7528\u6765\u5b58\u50a8\u4e00\u4e2a\u80fd\u91cf\u91cf\u5b50\u7684\u7a7a\u95f4\u3002\u5982\u679c\u67d0\u4e2a\u7a7a\u95f4\u5df2\u6ee1\uff0c\u5219\u8868\u793a\u5bf9\u5e94\u7684\u539f\u5b50\u5df2\u7ecf\u62e5\u6709\u4e86\u5b83\u7684\u80fd\u91cf\u91cf\u5b50\u3002<\/p>\n<p><center><br \/>\n<img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-zJa7H7iMOGY\/YF3QhG94SoI\/AAAAAAAAEuI\/olfdSHfeJOgKea3vHXEAyr5QCJ1ZjmcpgCLcBGAsYHQ\/s839\/cuantodeenergia.PNG\" \n     class=\"aligncenter lazyload\" alt=\"\u70ed\u529b\u5b66\u4e2d\u7684\u7ec4\u5408\u95ee\u9898\" width=\"400\" height=\"200\" \/><br \/>\n<\/center><\/p>\n<p>\u4e3a\u4e86\u8ba1\u7b97 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u4e2a\u80fd\u91cf\u91cf\u5b50\u5206\u5e03\u5230 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u4e2a\u7a7a\u95f4\u4e2d\u7684\u65b9\u5f0f\u6570\u91cf\uff0c\u6211\u4eec\u4f7f\u7528\u7ec4\u5408\u6570\u516c\u5f0f\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\binom{n}{k}=\\dfrac{n!}{k!(n-k)!}<\/span><\/span><\/p>\n<p>\u6b64\u516c\u5f0f\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u53ef\u80fd\u72b6\u6001\u6570\u91cf <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span>\u3002<\/p>\n<p><strong>(a)<\/strong> \u5982\u679c\u6709 10 \u4e2a\u91cf\u5b50\u5206\u5e03\u5230 10 \u4e2a\u7a7a\u95f4\u4e2d\uff0c\u90a3\u4e48\u53ea\u6709\u4e00\u79cd\u5206\u5e03\u65b9\u5f0f\u3002\u56e0\u6b64\uff0c<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega=1<\/span><\/span>\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\Omega = \\binom{10}{10}=\\dfrac{10!}{10!(10-10)!} = \\dfrac{10!}{10!0!} = 1 <\/span><\/span><\/p>\n<p><strong>(b)<\/strong> \u5bf9\u4e8e 5 \u4e2a\u91cf\u5b50\u5206\u5e03\u5230 10 \u4e2a\u7a7a\u95f4\uff0c\u6211\u4eec\u8fdb\u884c\u5982\u4e0b\u8ba1\u7b97\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\Omega &amp;= \\displaystyle\\binom{10}{5} \\\\ \\\\\n\n&amp;=\\dfrac{10!}{5!(10-5)!} = \\dfrac{10!}{5!\\cdot 5!} \\\\ \\\\\n\n&amp;= \\dfrac{5! \\cdot 6\\cdot 7\\cdot 8 \\cdot 9\\cdot 10}{5! \\cdot 2\\cdot 3\\cdot 4\\cdot 5} \\\\ \\\\\n\n&amp;= \\dfrac{ 7\\cdot 8 \\cdot 9\\cdot 10}{ 4\\cdot 5} = 7\\cdot 2 \\cdot 9 \\cdot 2 = 252\n\n\\end{array}<\/span>\n<p>\u56e0\u6b64\uff0c\u603b\u5171\u6709 252 \u79cd\u53ef\u80fd\u7684\u914d\u7f6e\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u6d89\u53ca\u5927\u6570\u7684\u95ee\u9898<\/h2>\n<p>\u6211\u4eec\u521a\u624d\u5206\u6790\u7684\u5185\u5bb9\u53ea\u662f\u4e00\u4e2a\u5f00\u59cb\u3002\u5982\u679c\u6211\u4eec\u5c06\u6848\u4f8b (b) \u7684\u7cfb\u7edf\u6269\u5c55\u5230\u5305\u542b 100 \u4e2a\u539f\u5b50\u548c 50 \u4e2a\u80fd\u91cf\u91cf\u5b50\uff0c\u6211\u4eec\u4f1a\u5f97\u5230 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega \\approx 10^{28}<\/span><\/span>\u3002\u73b0\u5728\uff0c\u8bd5\u60f3\u5982\u679c\u6211\u4eec\u7528\u4e00\u4e2a\u6469\u5c14\u7684\u539f\u5b50\u8fdb\u884c\u540c\u6837\u7684\u8ba1\u7b97\uff0c\u5176\u7ed3\u679c\u5c06\u662f\u96be\u4ee5\u60f3\u8c61\u7684\u3002<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>\u4f7f\u7528\u5bf9\u6570\u548c\u65af\u7279\u6797\u516c\u5f0f\u8fdb\u884c\u6570\u91cf\u7ea7\u8ba1\u7b97<\/h3>\n<p>\u5f53\u6211\u4eec\u5e0c\u671b\u4f30\u7b97\u5f62\u5982 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega = \\binom{n}{k}<\/span><\/span> \u7684\u6570\u91cf\u7ea7\uff0c\u5c24\u5176\u662f\u5728 <span class=\"katex-eq\" data-katex-display=\"false\">k=n\/2<\/span> \u7684\u60c5\u51b5\u4e0b\uff08\u6b64\u65f6\u8fbe\u5230\u6700\u5927\u503c\uff09\uff0c\u4f7f\u7528\u65af\u7279\u6797\u516c\u5f0f\u7684\u5bf9\u6570\u8fd1\u4f3c\u6cd5\u975e\u5e38\u6709\u7528\u3002<\/p>\n<p>\u4e3a\u4e86\u5904\u7406\u5982\u6b64\u5927\u7684\u6570\u5b57\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u53d6\u5bf9\u6570\u91cd\u65b0\u8868\u8ff0\u8ba1\u7b97\u8fc7\u7a0b\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\ln(\\Omega)=\\ln\\left(\\dfrac{n!}{k!(n-k)!}\\right)= \\ln(n!) - \\ln((n-k)!) - \\ln(k!)<\/span><\/span><\/p>\n<p>\u8fd9\u4e2a\u8868\u8fbe\u5f0f\u53ef\u4ee5\u901a\u8fc7\u65af\u7279\u6797\u516c\u5f0f\u7684\u9636\u4e58\u5bf9\u6570\u8fd1\u4f3c\u6765\u5904\u7406\uff0c\u5176\u4e2d\u6709\u4e24\u79cd\u53ef\u80fd\u7684\u7248\u672c\uff1a\u5e38\u89c4\u8fd1\u4f3c\u548c\u7b80\u5316\u8fd1\u4f3c\uff1a<\/p>\n<ul>\n<li><strong>\u5e38\u89c4\u8fd1\u4f3c\uff1a<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln(n!) \\approx \\dfrac{1}{2}\\ln(2n\\pi) + n\\ln(n) - n<\/span><\/span><\/li>\n<li><strong>\u7b80\u5316\u8fd1\u4f3c\uff1a<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln(n!) \\approx n\\ln(n) - n<\/span><\/span><\/li>\n<\/ul>\n<p><a name=\"4\"><\/a><\/p>\n<h4>\u901a\u8fc7\u7b80\u5316\u8fd1\u4f3c\u6cd5\u5c55\u5f00<\/h4>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/fnt99FeeHPM?si=H6r-5GhO2yqaeaXd\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><\/p>\n<p>\u4f7f\u7528\u7b80\u5316\u8fd1\u4f3c\u6cd5\uff0c\u53ef\u4ee5\u5f97\u5230\u4ee5\u4e0b\u7ed3\u679c\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\ln(\\Omega) &amp; \\approx n\\ln(n) - n - (n-k)\\ln(n-k) + (n-k) - k\\ln(k) + k \\\\ \\\\\n\n&amp;= n\\ln(n) - (n-k)\\ln(n-k) - k\\ln(k) \\\\ \\\\\n\n&amp;= n\\ln(n) - n\\ln(n-k) + k\\ln(n-k) - k\\ln(k) \\\\ \\\\\n\n&amp;= \\ln\\left[ \\left( \\dfrac{n}{n-k} \\right)^n \\right] + k\\ln\\left( \\dfrac{n-k}{k} \\right) \\\\ \\\\\n\n&amp;= \\ln\\left[ \\dfrac{1}{\\left(1 - \\dfrac{k}{n} \\right)^n} \\right] + k\\ln\\left( \\dfrac{n}{k} - 1 \\right)\n\n\\end{array}<\/span>\n<p>\u8003\u8651\u5230 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u503c\u8f83\u5927\uff0c\u53ef\u4ee5\u5e94\u7528\u4ee5\u4e0b\u5173\u7cfb\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{n\\to\\infty} \\left(1-\\dfrac{k}{n} \\right)^n = e^{-k} <\/span><\/span><\/p>\n<p>\u56e0\u6b64\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln(\\Omega) \\approx \\ln(e^k) + k\\ln\\left( \\dfrac{n}{k} -1 \\right) = k + k\\ln\\left( \\dfrac{n}{k} -1 \\right) <\/span><\/span><\/p>\n<p>\u6700\u7ec8\uff0c\u901a\u8fc7\u5bf9\u6570\u6362\u5e95\uff0c\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\log(\\Omega) = \\log(e)\\ln(\\Omega) \\approx k\\log(e)\\left[1 + \\ln\\left( \\dfrac{n}{k} - 1 \\right) \\right] <\/span><\/span><\/p>\n<p>\u7ed3\u679c\u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\boxed{\\Omega \\approx 10^{k\\log(e)\\left[1 + \\ln\\left( \\dfrac{n}{k} - 1 \\right) \\right]}}<\/span><\/span><\/p>\n<p>\u867d\u7136\u8fd9\u4e00\u7ed3\u679c\u672a\u63d0\u4f9b <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> \u7684\u786e\u5207\u503c\uff0c\u4f46\u5b83\u53ef\u4ee5\u4f30\u7b97\u51fa\u9700\u8981\u591a\u5c11\u4f4d\u6570\u6765\u8868\u793a\u7ed3\u679c\uff0c\u5e76\u4e14\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u53d8\u5f97\u66f4\u5927\u65f6\u7cbe\u786e\u5ea6\u4e5f\u4f1a\u63d0\u9ad8\u3002<\/p>\n<p>\u6b64\u5916\uff0c\u8be5\u65b9\u6cd5\u80fd\u5feb\u901f\u4f30\u7b97\u51fa <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> \u7684\u6700\u5927\u503c\u3002\u5f53 <span class=\"katex-eq\" data-katex-display=\"false\">k=n\/2<\/span> \u65f6\uff0c\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\text{Max}\\left(\\Omega\\right) \\approx 10^{\\dfrac{n}{2}\\log(e)\\left[1 + \\ln\\left( \\dfrac{n}{n\/2} - 1 \\right) \\right]} = 10^{ n\\log(e)\/2 } <\/span><\/span><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h4>\u901a\u8fc7\u5e38\u89c4\u8fd1\u4f3c\u6cd5\u5c55\u5f00<\/h4>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/i3OO4lHV24Q?si=jEPCafXqgVxNYqH-\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><\/p>\n<p>\u867d\u7136\u901a\u8fc7\u5e38\u89c4\u8fd1\u4f3c\u6cd5\u5c55\u5f00\u53ef\u4ee5\u5f97\u5230\u66f4\u7cbe\u786e\u7684\u7ed3\u679c\uff0c\u4f46\u5b83\u4f1a\u6d89\u53ca\u989d\u5916\u7684\u8ba1\u7b97\u6b65\u9aa4\u3002\u7136\u800c\uff0c\u5bf9\u4e8e\u8f83\u5927\u7684 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u503c\uff0c\u8fd9\u4e9b\u7ed3\u679c\u4e0e\u7b80\u5316\u8fd1\u4f3c\u6cd5\u7684\u7ed3\u679c\u5927\u81f4\u76f8\u540c\u3002\u4ee5\u4e0b\u662f\u57fa\u4e8e\u7b80\u5316\u8fd1\u4f3c\u6cd5\u7684\u57fa\u7840\u8fdb\u884c\u7684\u63a8\u5bfc\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rcl}\n\n\\ln(\\Omega) &amp; = &amp; \\ln\\left(\\dfrac{n!}{k!(n-k)!}\\right)= \\ln(n!) - \\ln((n-k)!) - \\ln(k!) \\\\ \\\\\n\n&amp; \\approx &amp; \\color{red}\\dfrac{1}{2}\\ln(2n\\pi)\\color{black} + n\\ln(n) - n \\\\ \\\\\n\n&amp; &amp; \\color{red}-\\dfrac{1}{2}\\ln(2(n-k)\\pi)\\color{black} - (n-k)\\ln(n-k) + (n-k) \\\\ \\\\\n\n&amp; &amp; \\color{red}-\\dfrac{1}{2}\\ln(2k\\pi)\\color{black} - k\\ln(k) + k\n\n\\end{array}<\/span>\n<p>\u7ea2\u8272\u90e8\u5206\u662f\u5e38\u89c4\u8fd1\u4f3c\u6cd5\u4e2d\u989d\u5916\u8003\u8651\u7684\u9879\uff0c\u5176\u4f59\u90e8\u5206\u4e0e\u7b80\u5316\u8fd1\u4f3c\u6cd5\u7684\u7ed3\u679c\u4e00\u81f4\u3002\u7531\u6b64\u53ef\u5f97\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rcl}\n\n\\ln(\\Omega) &amp; \\approx &amp; \\color{red}\\dfrac{1}{2}\\ln\\left( \\dfrac{2n\\pi}{2(n-k)\\pi \\cdot 2k\\pi} \\right)\\color{black} + k + k\\ln\\left(\\dfrac{n}{k} - 1\\right) \\\\ \\\\\n\n&amp; = &amp;  k + k\\ln\\left(\\dfrac{n}{k} - 1\\right) - \\dfrac{1}{2}\\ln\\left(\\dfrac{2k\\pi(n-k)}{n}\\right)\n\n\\end{array}<\/span>\n<p>\u7136\u540e\uff0c\u4f7f\u7528\u5bf9\u6570\u6362\u5e95\u6cd5\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\log(\\Omega) = \\log(e)\\ln(\\Omega) \\approx \\log(e) \\left[ k + k\\ln\\left(\\dfrac{n}{k} - 1\\right) - \\dfrac{1}{2}\\ln\\left(\\dfrac{2k\\pi(n-k)}{n}\\right) \\right] <\/span><\/span><\/p>\n<p>\u6700\u7ec8\uff0c\u53d6\u4ee5 10 \u4e3a\u5e95\u7684\u6307\u6570\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\Omega \\approx 10^{\\log(e) \\left[ k + k\\ln\\left(\\dfrac{n}{k} - 1\\right) - \\dfrac{1}{2}\\ln\\left(\\dfrac{2k\\pi(n-k)}{n}\\right) \\right]}\n\n<\/span>\n<p>\u7c7b\u4f3c\u4e8e\u4e4b\u524d\u7684\u60c5\u51b5\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5c06 <span class=\"katex-eq\" data-katex-display=\"false\">k=n\/2<\/span> \u4ee3\u5165\u6765\u627e\u5230\u6b64\u6570\u7684\u6700\u5927\u503c\uff0c\u7ed3\u679c\u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rcl}\n\n\\text{Max}(\\Omega) &amp;\\approx &amp; 10^{\\log(e) \\left[ \\dfrac{n}{2} + \\dfrac{n}{2}\\ln\\left(\\dfrac{n}{(n\/2)} - 1\\right) - \\dfrac{1}{2}\\ln\\left(\\dfrac{2(n\/2)\\pi(n-n\/2)}{n}\\right) \\right]} \\\\ \\\\\n\n&amp; = &amp; 10^{\\log(e) \\left[\\dfrac{n}{2} - \\dfrac{1}{2}\\ln\\left(\\dfrac{n\\pi}{2} \\right) \\right]} = 10^{\\log(e)(n-\\ln(n\\pi\/2))\/2}\n\n\\end{array}<\/span>\n<p><a name=\"6\"><\/a><\/p>\n<h2>\u7ec4\u5408\u8ba1\u7b97\u4e0e\u6570\u91cf\u7ea7\u8ba1\u7b97\u793a\u4f8b<\/h2>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/M7NrtICrSzU?si=6UT34333Wp5YwFU1\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>\u6848\u4f8b 1: \u5927\u9636\u4e58<\/h3>\n<p>\u4f30\u7b97 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(10^{50}\\right)!<\/span><\/span> \u7684\u6570\u91cf\u7ea7\uff0c\u5373\u4e66\u5199\u8be5\u6570\u503c\u6240\u9700\u7684\u4f4d\u6570\u3002<\/p>\n<h5>\u89e3\u7b54<\/h5>\n<p>\u4e3a\u4e86\u8fdb\u884c\u8be5\u8ba1\u7b97\uff0c\u6211\u4eec\u4f7f\u7528\u65af\u7279\u6797\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\ln\\left[ \\left(10^{50}\\right)! \\right] &amp;\\approx 10^{50}\\ln\\left(10^{50}\\right) - 10^{50}\\\\ \\\\\n\n&amp;= \\left[\\ln\\left(10^{50}\\right) -1\\right]10^{50}  \\\\ \\\\\n\n&amp;= \\left[50\\ln(10)-1 \\right]10^{50} \\\\ \\\\\n\n\\end{array}\n\n<\/span>\n<p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5e94\u7528\u5bf9\u6570\u6362\u5e95\u516c\u5f0f\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\ln\\left[ \\left(10^{50}\\right)! \\right] = \\dfrac{\\log\\left[\\left(10^{50}\\right)!\\right]}{\\log{e}}<\/span>\n<p>\u56e0\u6b64\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\log\\left[ \\left(10^{50}\\right)! \\right] \\approx \\log(e)\\left[50\\ln(10)-1 \\right]10^{50}\n\n<\/span>\n<p>\u6700\u540e\uff0c\u53d6\u4ee5 10 \u4e3a\u5e95\u7684\u6307\u6570\uff0c\u5f97\u5230\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\left(10^{50}\\right)!  \\approx  10^{\\log(e)\\left[50\\ln(10)-1 \\right]10^{50}} = 10^{49,5657 \\cdot 10^{50}}\n\n<\/span>\n<p>\u8be5\u6570\u7684\u6570\u91cf\u7ea7\u7531\u5176\u4ee5 10 \u4e3a\u5e95\u7684\u6307\u6570\u51b3\u5b9a\uff0c\u8868\u660e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(10^{50}\\right)!<\/span><\/span> \u662f\u4e00\u4e2a\u975e\u5e38\u5e9e\u5927\u7684\u6570\u5b57\u3002<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h3>\u6848\u4f8b 2: \u5927\u7ec4\u5408<\/h3>\n<p>\u4e00\u6240\u666e\u901a\u4f4f\u5b85\u7ea6\u6709 12 \u4e2a\u7535\u706f\u5f00\u5173\uff0c\u8fd9\u4e9b\u5f00\u5173\u53ef\u4ee5\u5904\u4e8e\u6253\u5f00\u6216\u5173\u95ed\u72b6\u6001\u3002\u5e73\u5747\u6bcf\u6240\u4f4f\u5b85\u6709 4 \u4e2a\u4eba\u3002\u5982\u679c\u4e00\u5ea7\u57ce\u5e02\u6709 500 \u4e07\u4eba\u53e3\uff0c\u95ee\u8fd9\u5ea7\u57ce\u5e02\u4e2d\u6709\u4e00\u534a\u5f00\u5173\u6253\u5f00\u7684\u6240\u6709\u53ef\u80fd\u60c5\u51b5\u6709\u591a\u5c11\u79cd\uff1f<\/p>\n<h5>\u89e3\u7b54<\/h5>\n<p>\u57ce\u5e02\u4e2d\u603b\u5f00\u5173\u6570 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u4e3a\uff1a<\/p>\n<p style=\"text-align:center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rcl}\n\nn &amp;=&amp;\\dfrac{\\text{\u57ce\u5e02\u603b\u4eba\u53e3}}{\\text{\u6bcf\u6237\u4eba\u6570}} \\times \\text{\u6bcf\u6237\u5f00\u5173\u6570} \\\\ \\\\\n\n&amp;=&amp; \\dfrac{5\\cdot 10^6}{4}\\cdot 12 = 15\\cdot 10^6\n\n\\end{array}\n\n<\/span>\n<p>\u6240\u6709\u5f00\u5173\u4e2d\u4e00\u534a\u6253\u5f00\u65f6\u7684\u5fae\u89c2\u6001\u6570\u76ee <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{max}<\/span>\uff0c\u53ef\u4ee5\u901a\u8fc7\u4e24\u79cd\u65b9\u6cd5\u4f30\u7b97\uff1a<\/p>\n<ul>\n<ol><strong>\u5e38\u89c4\u8fd1\u4f3c\u4f30\u7b97\uff1a<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{max} = 10^{\\log(e)\\left[15\\cdot10^6 - \\ln\\left(15\\pi\\cdot10^6 \/ 2 \\right) \\right]\/2} \\approx 10^{6.514.413,542}<\/span><\/span><\/ol>\n<ol><strong>\u7b80\u5316\u8fd1\u4f3c\u4f30\u7b97\uff1a<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{max} = 10^{\\log(e)\\left[15\\cdot10^6 \\right]\/2} \\approx 10^{6.514.417,229}<\/span><\/span><\/ol>\n<\/ul>\n<p>\u867d\u7136\u4e24\u79cd\u8fd1\u4f3c\u7ed3\u679c\u4e4b\u95f4\u76f8\u5dee\u7ea6 4 \u4e2a\u6570\u91cf\u7ea7\uff08\u770b\u4f3c\u5de8\u5927\uff09\uff0c\u4f46\u4e0e\u8d85\u8fc7 650 \u4e07\u6570\u91cf\u7ea7\u7684\u89c4\u6a21\u76f8\u6bd4\uff0c\u8fd9\u70b9\u5dee\u8ddd\u663e\u5f97\u5fae\u4e0d\u8db3\u9053\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u70ed\u529b\u5b66\u4e2d\u7684\u7ec4\u5408\u95ee\u9898 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