{"id":30710,"date":"2021-05-06T13:00:45","date_gmt":"2021-05-06T13:00:45","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30710"},"modified":"2025-01-01T06:04:12","modified_gmt":"2025-01-01T06:04:12","slug":"%e7%83%ad%e5%8a%9b%e5%ad%a6%e4%b8%ad%e7%9a%84%e5%be%ae%e8%a7%82%e6%80%81%e4%b8%8e%e5%ae%8f%e8%a7%82%e6%80%81","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%e5%be%ae%e8%a7%82%e6%80%81%e4%b8%8e%e5%ae%8f%e8%a7%82%e6%80%81\/","title":{"rendered":"\u70ed\u529b\u5b66\u4e2d\u7684\u5fae\u89c2\u6001\u4e0e\u5b8f\u89c2\u6001"},"content":{"rendered":"<style>\n\tp, ul, ol {\n\t\ttext-align: justify;\n\t}\n\th1, h2 {\n\t\ttext-align:center;\n\t}\n<\/style>\n<h1>\u5fae\u89c2\u72b6\u6001\u4e0e\u5b8f\u89c2\u72b6\u6001\u5728\u70ed\u529b\u5b66\u4e2d\u7684\u610f\u4e49<\/h1>\n<p style=\"text-align:center;\"><em>\u5728\u672c\u8282\u8bfe\u4e2d\uff0c\u4f60\u5c06\u5b66\u4e60\u5982\u4f55\u533a\u5206\u70ed\u529b\u5b66\u4e2d\u7684\u5fae\u89c2\u72b6\u6001\u548c\u5b8f\u89c2\u72b6\u6001\uff0c\u63a2\u7d22\u7cfb\u7edf\u7684\u5fae\u89c2\u914d\u7f6e\u5982\u4f55\u51b3\u5b9a\u5176\u5b8f\u89c2\u6027\u8d28\u3002\u901a\u8fc7\u5177\u4f53\u7684\u4f8b\u5b50\uff0c\u4f8b\u5982\u5728\u5177\u6709\u591a\u4e2a\u53ef\u80fd\u72b6\u6001\u7684\u7cfb\u7edf\u4e2d\u8ba1\u7b97\u7ec4\u5408\u6570\uff0c\u4f60\u5c06\u7406\u89e3\u8fde\u63a5\u5fae\u89c2\u4e0e\u53ef\u89c2\u6d4b\u73b0\u8c61\u7684\u5173\u952e\u6982\u5ff5\u3002\u4e86\u89e3\u8fd9\u4e9b\u601d\u60f3\u5982\u4f55\u6210\u4e3a\u89e3\u91ca\u590d\u6742\u70ed\u7cfb\u7edf\u884c\u4e3a\u7684\u57fa\u7840\uff0c\u5e76\u57f9\u517b\u5bf9\u652f\u914d\u7269\u8d28\u7684\u539f\u5219\u7684\u65b0\u89c6\u89d2\u3002<\/em><\/p>\n<p style=\"text-align:center;\"><strong>\u5b66\u4e60\u76ee\u6807\uff1a<\/strong><br \/>\n\u5b8c\u6210\u672c\u8282\u8bfe\u540e\uff0c\u5b66\u751f\u5c06\u80fd\u591f\uff1a\n<\/p>\n<ol>\n<li><strong>\u533a\u5206<\/strong>\u5fae\u89c2\u72b6\u6001\u548c\u5b8f\u89c2\u72b6\u6001\uff0c\u7406\u89e3\u5176\u5b9a\u4e49\u53ca\u5176\u5728\u70ed\u529b\u5b66\u4e2d\u7684\u5e94\u7528\u3002<\/li>\n<li><strong>\u5206\u7c7b<\/strong>\u5fae\u89c2\u72b6\u6001\u4e3a\u5b8f\u89c2\u72b6\u6001\uff0c\u6839\u636e\u5171\u540c\u7279\u6027\u5e76\u901a\u8fc7\u7ec4\u5408\u8ba1\u7b97\u5176\u6570\u91cf\u3002<\/li>\n<li><strong>\u89e3\u91ca<\/strong>\u5fae\u89c2\u72b6\u6001\u7684\u7b49\u53ef\u80fd\u6027\u5982\u4f55\u51b3\u5b9a\u70ed\u529b\u5b66\u7cfb\u7edf\u4e2d\u5b8f\u89c2\u72b6\u6001\u7684\u6982\u7387\u3002<\/li>\n<li><strong>\u5173\u8054<\/strong>\u7cfb\u7edf\u7684\u5fae\u89c2\u6027\u8d28\u4e0e\u5176\u5b8f\u89c2\u6027\u8d28\uff0c\u4f8b\u5982\u538b\u529b\u3001\u4f53\u79ef\u548c\u80fd\u91cf\u3002<\/li>\n<li><strong>\u5206\u6790<\/strong>\u901a\u8fc7\u5b9e\u9645\u4f8b\u5b50\u7814\u7a76\u70ed\u7cfb\u7edf\uff0c\u5f3a\u8c03\u5fae\u89c2\u72b6\u6001\u548c\u5b8f\u89c2\u72b6\u6001\u7684\u91cd\u8981\u6027\u3002<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>\u5185\u5bb9\u76ee\u5f55<\/u>\uff1a<\/strong><br \/>\n<a href=\"#1\">\u70ed\u529b\u5b66\u4e2d\u5fae\u89c2\u72b6\u6001\u548c\u5b8f\u89c2\u72b6\u6001\u7684\u533a\u522b<\/a><br \/>\n<a href=\"#2\">\u70ed\u529b\u5b66\u4e2d\u5b8f\u89c2\u72b6\u6001\u7684\u7279\u6027<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/O5RKp4Dt8Z0\" title=\"YouTube \u89c6\u9891\u64ad\u653e\u5668\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>\u70ed\u529b\u5b66\u4e2d\u5fae\u89c2\u72b6\u6001\u548c\u5b8f\u89c2\u72b6\u6001\u7684\u533a\u522b<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=O5RKp4Dt8Z0&amp;t=7s\" target=\"_blank\" rel=\"noopener\"><strong>\u5728\u70ed\u529b\u5b66\u4e2d\u4e00\u4e2a\u91cd\u8981\u7684\u533a\u5206<\/strong><\/a>\u662f\u5fae\u89c2\u72b6\u6001\u548c\u5b8f\u89c2\u72b6\u6001\u7684\u533a\u522b\u3002\u901a\u5e38\u8ba4\u4e3a\u5fae\u89c2\u72b6\u6001\u4e0e\u7269\u8d28\u7684\u5fae\u89c2\u7279\u6027\u6709\u5173\uff0c\u800c\u5b8f\u89c2\u72b6\u6001\u5219\u6d89\u53ca\u7531\u8bb8\u591a\u5fae\u89c2\u7cfb\u7edf\u7ec4\u6210\u7684\u6574\u4f53\u7cfb\u7edf\u3002\u867d\u7136\u8fd9\u79cd\u533a\u5206\u6709\u52a9\u4e8e\u521d\u6b65\u7406\u89e3\u8fd9\u4e9b\u6982\u5ff5\uff0c\u4f46\u4ec5\u51ed\u8fd9\u4e00\u70b9\u5e76\u4e0d\u8db3\u4ee5\u5b8c\u5168\u6db5\u76d6\u5fae\u89c2\u72b6\u6001\u548c\u5b8f\u89c2\u72b6\u6001\u7684\u672c\u8d28\u542b\u4e49\u3002<\/p>\n<p>\u4e3a\u4e86\u66f4\u51c6\u786e\u5730\u7406\u89e3\u8fd9\u4e9b\u6982\u5ff5\uff0c\u8ba9\u6211\u4eec\u901a\u8fc7\u4ee5\u4e0b\u4f8b\u5b50\u8fdb\u884c\u8bf4\u660e\uff1a<\/p>\n<h3>\u7528\u786c\u5e01\u89e3\u91ca\u5fae\u89c2\u72b6\u6001\u4e0e\u5b8f\u89c2\u72b6\u6001<\/h3>\n<p><strong><a href=\"https:\/\/www.youtube.com\/watch?v=O5RKp4Dt8Z0&amp;t=218s\" target=\"_blank\" rel=\"noopener\">\u5047\u8bbe\u4f60\u6709\u4e00\u4e2a\u88c5\u6709100\u679a\u786c\u5e01\u7684\u76d2\u5b50\uff0c<\/a><\/strong>\u76d2\u5b50\u59cb\u7ec8\u5904\u4e8e\u5173\u95ed\u72b6\u6001\u3002\u5982\u679c\u4f60\u7528\u529b\u6447\u6643\u76d2\u5b50\uff0c\u786c\u5e01\u5728\u5185\u90e8\u8df3\u52a8\u5e76\u843d\u5230\u67d0\u79cd\u72b6\u6001\uff1a\u6b63\u9762\u6216\u53cd\u9762\u3002\u6574\u4e2a\u7cfb\u7edf\u53ef\u80fd\u7684\u72b6\u6001\u603b\u6570\u4e3a<span class=\"katex-eq\" data-katex-display=\"false\">2^{100}<\/span>\uff08\u8bd5\u7740\u4f30\u7b97\u8fd9\u4e2a\u6570\u5b57\u7684\u6570\u91cf\u7ea7\uff01\uff09\uff0c\u6bcf\u79cd\u72b6\u6001\u5bf9\u5e94\u4e0d\u540c\u7684\u6b63\u53cd\u9762\u914d\u7f6e\u3002\u5047\u8bbe\u6bcf\u79cd\u914d\u7f6e\u7684\u53ef\u80fd\u6027\u662f\u76f8\u7b49\u7684\uff0c\u56e0\u6b64\u6bcf\u79cd\u914d\u7f6e\u53d1\u751f\u7684\u6982\u7387\u662f<span class=\"katex-eq\" data-katex-display=\"false\">1\/2^{100}<\/span>\u3002<\/p>\n<h4>\u6bcf\u4e00\u79cd\u7279\u5b9a\u7684\u914d\u7f6e\u5c31\u662f\u4e00\u4e2a\u5fae\u89c2\u72b6\u6001<\/h4>\n<p>\u6211\u4eec\u53ef\u4ee5\u8bf4\uff0c\u6bcf\u4e00\u79cd\u7279\u5b9a\u7684\u914d\u7f6e\u662f\u7cfb\u7edf\u7684\u4e00\u4e2a\u5fae\u89c2\u72b6\u6001\u3002\u4f8b\u5982\uff0c\u7b2c\u4e00\u679a\u786c\u5e01\u662f\u6b63\u9762\uff0c\u7b2c\u4e8c\u679a\u662f\u53cd\u9762\uff0c\u7b2c\u4e09\u679a\u662f\u6b63\u9762\uff0c\u4f9d\u6b64\u7c7b\u63a8\u3002\u8bc6\u522b\u4e00\u4e2a\u5fae\u89c2\u72b6\u6001\u9700\u8981\u5355\u72ec\u6807\u8bc6\u6bcf\u4e00\u679a\u786c\u5e01\uff0c\u5728\u5b9e\u9645\u64cd\u4f5c\u4e2d\uff0c\u8fd9\u79cd\u65b9\u6cd5\u975e\u5e38\u7e41\u7410\u3002<\/p>\n<h4>\u5b8f\u89c2\u72b6\u6001\u662f\u5177\u6709\u5171\u540c\u7279\u6027\u7684\u6240\u6709\u5fae\u89c2\u72b6\u6001\u7684\u96c6\u5408<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=O5RKp4Dt8Z0&amp;t=372s\" target=\"_blank\" rel=\"noopener\"><strong>\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u53ef\u4ee5\u9009\u62e9\u4e0d\u9010\u4e00\u7814\u7a76<\/strong><\/a>\u6bcf\u4e2a\u5fae\u89c2\u72b6\u6001\uff0c\u800c\u662f\u5c06\u6240\u6709\u53ef\u80fd\u7684\u5fae\u89c2\u72b6\u6001\u5206\u7c7b\uff0c\u6839\u636e\u5b83\u4eec\u7684\u5171\u540c\u7279\u6027\uff0c\u4f8b\u5982\u5177\u6709\u4e00\u4e2a\u6b63\u9762\u3001\u4e24\u4e2a\u6b63\u9762\u3001\u4e09\u4e2a\u6b63\u9762\u7b49\u3002\u901a\u8fc7\u7ec4\u5408\u6570\u8ba1\u7b97\uff0c\u6211\u4eec\u5f97\u51fa\uff1a<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>0\u4e2a\u6b63\u9762\u548c100\u4e2a\u53cd\u9762\u7684\u72b6\u6001\u6570\uff1a<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle{{100}\\choose{0}} = \\frac{100!}{(100 - 0)! 0!} = 1 <\/span><\/li>\n<li>1\u4e2a\u6b63\u9762\u548c99\u4e2a\u53cd\u9762\u7684\u72b6\u6001\u6570\uff1a<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle {{100}\\choose{1}} = \\frac{100!}{(100 - 1)! 1!}= \\frac{100!}{99!} = 100 <\/span><\/li>\n<li>2\u4e2a\u6b63\u9762\u548c98\u4e2a\u53cd\u9762\u7684\u72b6\u6001\u6570\uff1a<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle {{100}\\choose{2}} = \\frac{100!}{(100 - 2)! 2!}= \\frac{100!}{98!2!} = \\frac{99\\cdot 100}{2} = 99 \\cdot 50 = 4950 <\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li>49\u4e2a\u6b63\u9762\u548c51\u4e2a\u53cd\u9762\u7684\u72b6\u6001\u6570\uff1a<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle{{100}\\choose{49}} = \\frac{100!}{(100 - 49)! 49!}= \\frac{100!}{51!49!}\\approx 9.89\\cdot 10^{28}<\/span><\/li>\n<li>50\u4e2a\u6b63\u9762\u548c50\u4e2a\u53cd\u9762\u7684\u72b6\u6001\u6570\uff1a<span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle{{100}\\choose{50}} = \\frac{100!}{(100 - 50)! 50!}= \\frac{100!}{50!50!}\\approx 10^{29}<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>\u56e0\u6b64\uff0c\u5b8f\u89c2\u72b6\u6001\u201c0\u4e2a\u6b63\u9762\u201d\u5305\u542b1\u4e2a\u5fae\u89c2\u72b6\u6001\uff0c\u201c1\u4e2a\u6b63\u9762\u201d\u5305\u542b100\u4e2a\u5fae\u89c2\u72b6\u6001\uff0c\u201c2\u4e2a\u6b63\u9762\u201d\u5305\u542b4950\u4e2a\u5fae\u89c2\u72b6\u6001\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>\u70ed\u529b\u5b66\u4e2d\u5b8f\u89c2\u72b6\u6001\u7684\u7279\u6027<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=O5RKp4Dt8Z0&amp;t=607s\" target=\"_blank\" rel=\"noopener\"><strong>\u4e00\u822c\u6765\u8bf4\uff0c\u4e00\u4e2a\u5b8f\u89c2\u72b6\u6001\u7531\u8bb8\u591a\u7b49\u53ef\u80fd\u7684\u5fae\u89c2\u72b6\u6001\u7ec4\u6210\u3002<\/strong><\/a><\/p>\n<p>\u867d\u7136\u6240\u6709\u7684\u5fae\u89c2\u72b6\u6001\u662f\u7b49\u53ef\u80fd\u7684\uff0c\u4f46\u5b8f\u89c2\u72b6\u6001\u901a\u5e38\u4e0d\u662f\u7b49\u53ef\u80fd\u7684\u3002\u5b8f\u89c2\u72b6\u6001\u7684\u6982\u7387\u4e0e\u5305\u542b\u7684\u5fae\u89c2\u72b6\u6001\u6570\u91cf\u6210\u6b63\u6bd4\u3002\u6700\u53ef\u80fd\u7684\u5b8f\u89c2\u72b6\u6001\u662f\u5305\u542b\u5fae\u89c2\u72b6\u6001\u6700\u591a\u7684\u72b6\u6001\u3002<\/p>\n<p>\u70ed\u7cfb\u7edf\u7684\u884c\u4e3a\u4e0e\u4e0a\u8ff0\u4f8b\u5b50\u975e\u5e38\u76f8\u4f3c\u3002\u8981\u63cf\u8ff0\u70ed\u529b\u5b66\u7cfb\u7edf\u4e2d\u7684\u5fae\u89c2\u72b6\u6001\uff0c\u9700\u8981\u6307\u660e\u5b9a\u4e49\u7cfb\u7edf\u5fae\u89c2\u914d\u7f6e\u7684\u91cf\uff0c\u4f8b\u5982\u539f\u5b50\u7684\u4f4d\u7f6e\u4fe1\u606f\u3001\u901f\u5ea6\u6216\u80fd\u91cf\u3002<\/p>\n<p>\u5b9e\u9645\u4e0a\uff0c\u5f88\u96be\u6d4b\u91cf\u7cfb\u7edf\u7684\u6bcf\u4e00\u4e2a\u5fae\u89c2\u72b6\u6001\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5b8f\u89c2\u72b6\u6001\u53ef\u4ee5\u7528\u7cfb\u7edf\u7684\u5b8f\u89c2\u5c5e\u6027\u6765\u63cf\u8ff0\uff0c\u5982\u538b\u529b\u3001\u603b\u80fd\u91cf\u6216\u4f53\u79ef\u3002\u4e00\u4e2a\u4f53\u79ef\u4e3a<span 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