{"id":2127,"date":"2023-04-02T09:42:02","date_gmt":"2023-04-02T01:42:02","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2127"},"modified":"2023-04-05T20:42:30","modified_gmt":"2023-04-05T12:42:30","slug":"fundamentals-of-high-school-mathematics-arranging-and-combining","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/02\/fundamentals-of-high-school-mathematics-arranging-and-combining\/","title":{"rendered":"\u9ad8\u4e2d\u6570\u5b66\u57fa\u7840\uff1a\u6392\u5217\u4e0e\u7ec4\u5408"},"content":{"rendered":"

\u57fa\u672c\u8ba1\u6570\u539f\u7406<\/h2>\n

\uff081\uff09\u5206\u7c7b\u52a0\u6cd5\u8ba1\u6570\u539f\u7406\uff08\u5206\u7c7b\u76f8\u52a0\uff09<\/p>\n

\u505a\u4e00\u4ef6\u4e8b\u60c5\uff0c\u5b8c\u6210\u5b83\u6709$n$\u7c7b\u529e\u6cd5\uff0c\u5728\u7b2c1\u7c7b\u529e\u6cd5\u4e2d\u6709$m_1$\u4e2d\u4e0d\u540c\u7684\u65b9\u6cd5\uff0c\u5728\u7b2c2\u7c7b\u529e\u6cd5\u4e2d\u6709$m_2$\u4e2d\u4e0d\u540c\u7684\u65b9\u6cd5\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u5728\u7b2cn\u7c7b\u529e\u6cd5\u4e2d\u6709$m_n$\u4e2d\u4e0d\u540c\u7684\u65b9\u6cd5\u3002\u90a3\u4e48\u5b8c\u6210\u8fd9\u4ef6\u4e8b\u60c5\u5171\u6709$N=m_1+m_2+\u00b7\u00b7\u00b7+m_n$\u79cd\u4e0d\u540c\u7684\u65b9\u6cd5\u3002<\/p>\n

<\/p>\n

\uff082\uff09\u5206\u7c7b\u4e58\u6cd5\u8ba1\u6570\u539f\u7406\uff08\u5206\u6b65\u76f8\u4e58\uff09<\/p>\n

\u505a\u4e00\u4ef6\u4e8b\u60c5\uff0c\u5b8c\u6210\u5b83\u9700\u8981$n$\u4e2a\u6b65\u9aa4\uff0c\u505a\u7b2c1\u4e2a\u6b65\u9aa4\u4e2d\u6709$m_1$\u4e0d\u540c\u7684\u65b9\u6cd5\uff0c\u505a\u7b2c2\u4e2a\u6b65\u9aa4\u4e2d\u6709$m_2$\u4e0d\u540c\u7684\u65b9\u6cd5\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7\u505a\u7b2cn\u4e2a\u6b65\u9aa4\u6709$m_n$\u4e2d\u4e0d\u540c\u7684\u65b9\u6cd5\u3002\u90a3\u4e48\u5b8c\u6210\u8fd9\u4ef6\u4e8b\u60c5\u5171\u6709$N=m_1\u00d7m_2\u00d7\u00b7\u00b7\u00b7\u00d7m_n$\u79cd\u4e0d\u540c\u7684\u65b9\u6cd5\u3002<\/p>\n

\u6392\u5217\u4e0e\u7ec4\u5408<\/h2>\n

\u6392\u5217<\/h3>\n

\u2460 \u4ece$n$\u4e2a\u4e0d\u540c\u5143\u7d20\u4e2d\uff0c\u4efb\u53d6$m(m\u2264n)$\u4e2a\u5143\u7d20\uff0c\u6309\u7167\u4e00\u5b9a\u7684\u987a\u5e8f\u6392\u6210\u4e00\u5217\uff0c\u53eb\u505a\u4ece$n$\u4e2a\u4e0d\u540c\u5143\u7d20\u4e2d\u53d6\u51fa$m$\u4e2a\u5143\u7d20\u7684\u4e00\u4e2a\u6392\u5217\u3002
\n\u2461 \u4ece$n$\u4e2a\u4e0d\u540c\u5143\u7d20\u4e2d\uff0c\u53d6\u51fa$m(m\u2264n$\u4e2a\u5143\u7d20\u7684\u6240\u6709\u6392\u5217\u7684\u4e2a\u6570\uff0c\u53eb\u505a\u4ece$n$\u4e2a\u4e0d\u540c\u5143\u7d20\u4e2d\u53d6\u51fa$m$\u4e2a\u5143\u7d20\u7684\u6392\u5217\u6570\uff0c\u8bb0\u4f5c$A_n^m$\u3002<\/p>\n

\u7ec4\u5408<\/h3>\n

\u2460 \u4ece$n$\u4e2a\u4e0d\u540c\u5143\u7d20\u4e2d\uff0c\u4efb\u53d6$m(m\u2264n)$\u4e2a\u5143\u7d20\u5e76\u6210\u4e00\u7ec4\uff0c\u53eb\u505a\u4ece$n$\u4e2a\u4e0d\u540c\u5143\u7d20\u4e2d\u53d6\u51fa$m$\u4e2a\u5143\u7d20\u7684\u4e00\u4e2a\u7ec4\u5408\u3002
\n\u2461 \u4ece$n$\u4e2a\u4e0d\u540c\u5143\u7d20\u4e2d\uff0c\u53d6\u51fa$m(m\u2264n)$\u4e2a\u5143\u7d20\u7684\u6240\u6709\u7ec4\u5408\u7684\u4e2a\u6570\uff0c\u53eb\u505a\u4ece$n$\u4e2a\u4e0d\u540c\u5143\u7d20\u4e2d\u53d6\u51fa$m$\u4e2a\u5143\u7d20\u7684\u7ec4\u5408\u6570\uff0c\u8bb0\u4f5c$C_n^m$\u3002<\/p>\n

\u6392\u5217\u6570\u516c\u5f0f<\/h3>\n

$
\nA_n^m=n(n-1)(n-2)\u00b7\u00b7\u00b7(n-m+1)\\
\n\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{n!}{(n-m)!}
\n$<\/p>\n

$A_n^n=n!$\uff0c\u89c4\u5b9a$0!=1$<\/p>\n

\u7ec4\u5408\u6570\u516c\u5f0f<\/h3>\n

$
\nC_n^m=\\frac{n(n-1)(n-2)\u00b7\u00b7\u00b7(n-m+1)}{m!}\\
\n\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{n!}{m!(n-m)!}
\n$<\/p>\n

$C_n^n=Cn^{n-m}$\uff0c\u89c4\u5b9a$C_n^0=1$<\/p>\n

\u7ec4\u5408\u8bf1\u5bfc\u516c\u5f0f<\/h3>\n

$Cn^m + C<\/em>{n-1}^m = C_n^{m+1}$<\/p>\n

$C_n^m + Cn^{m-1} = C<\/em>{n+1}^m$<\/p>\n

Python\u5b9e\u73b0<\/h3>\n
# -*- coding: utf-8 -*-\nimport itertools\n\nx = [1, 2, 3, 4, 5]\n# \u7ec4\u5408\nc1 = itertools.combinations(x, 3)\nprint(list(c1))\n# \u6392\u5217\na1 = itertools.permutations(x, 3)\nprint(list(a1))<\/code><\/pre>\n
\u6392\u5217\u4e0e\u7ec4\u5408\u7684\u533a\u522b<\/h3>\n
\u6392\u5217\u6709\u987a\u5e8f\uff0c\u7ec4\u5408\u65e0\u987a\u5e8f<\/p>\n
\u6392\u5217\u4e0e\u7ec4\u5408\u7684\u8054\u7cfb<\/h3>\n
$A_n^m=C_n^m\u00b7A_m^m$<\/p>\n
\u5373\u6392\u5217\u5c31\u662f\u5148\u7ec4\u5408\u5728\u5168\u6392\u5217<\/p>\n
\u6392\u5217\u4e0e\u7ec4\u5408\u7684\u4e24\u4e2a\u6027\u8d28<\/h3>\n
\u6392\u5217\uff1a$A_{n+1}^m=A_n^m+mAn^{m-1}$
\n\u7ec4\u5408\uff1a$C<\/em>{n+1}^m=C_n^m+C_n^{m-1}$<\/p>\n","protected":false},"excerpt":{"rendered":"
\u57fa\u672c\u8ba1\u6570\u539f\u7406 \uff081\uff09\u5206\u7c7b\u52a0\u6cd5\u8ba1\u6570\u539f\u7406\uff08\u5206\u7c7b\u76f8\u52a0\uff09 \u505a\u4e00\u4ef6\u4e8b\u60c5\uff0c\u5b8c\u6210\u5b83\u6709$n$\u7c7b\u529e\u6cd5\uff0c\u5728\u7b2c1\u7c7b\u529e\u6cd5\u4e2d\u6709$m_1$ […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[509],"tags":[],"class_list":["post-2127","post","type-post","status-publish","format-standard","hentry","category-mathematics-fundamentals"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2127","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2127"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2127\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2127"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2127"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2127"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}