{"id":57318,"date":"2014-09-19T01:08:19","date_gmt":"2014-09-18T17:08:19","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=57318"},"modified":"2021-10-06T16:15:15","modified_gmt":"2021-10-06T08:15:15","slug":"%e5%be%9e%e4%ba%8c%e9%a0%85%e5%bc%8f%e5%ae%9a%e7%90%86%e5%88%b0%e5%a4%9a%e9%a0%85%e5%bc%8f%e5%ae%9a%e7%90%862%ef%bc%88from-binomial-theorem-to-multinomial-theorem-2%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%e5%be%9e%e4%ba%8c%e9%a0%85%e5%bc%8f%e5%ae%9a%e7%90%86%e5%88%b0%e5%a4%9a%e9%a0%85%e5%bc%8f%e5%ae%9a%e7%90%862%ef%bc%88from-binomial-theorem-to-multinomial-theorem-2%ef%bc%89\/","title":{"rendered":"\u5f9e\u4e8c\u9805\u5f0f\u5b9a\u7406\u5230\u591a\u9805\u5f0f\u5b9a\u7406 (2)"},"content":{"rendered":"

\u5f9e\u4e8c\u9805\u5f0f\u5b9a\u7406\u5230\u591a\u9805\u5f0f\u5b9a\u7406 (2)\uff08From Binomial Theorem to Multinomial Theorem (2)\uff09<\/strong><\/span>
\n\u81fa\u5317\u5e02\u7acb\u7b2c\u4e00\u5973\u5b50\u9ad8\u7d1a\u4e2d\u5b78\u8607\u4fca\u9d3b\u8001\u5e2b<\/strong><\/span><\/p>\n

\u9023\u7d50\uff1a\u5f9e\u4e8c\u9805\u5f0f\u5b9a\u7406\u5230\u591a\u9805\u5f0f\u5b9a\u7406(1)<\/a><\/p>\n

\u5728\u3008\u5f9e\u4e8c\u9805\u5f0f\u5b9a\u7406\u5230\u591a\u9805\u5f0f\u5b9a\u7406(1)\u3009\u4e2d\u63d0\u5230 \\((x+y)^3\\)\u00a0\u7684 \\(x^2y^1\\)\u00a0\u9805\u662f\u5982\u4f55\u7522\u751f\u5462\uff1f\u7531\u65bc\u00a0\\({\\left( {x + y} \\right)^3} = \\left( {x + y} \\right)\\left( {x + y} \\right)\\left( {x + y} \\right)\\)\uff0c\u6545\u53ef\u770b\u6210\u5728\u4e09\u500b \\((x+y)\\)\u00a0\u62ec\u865f\u4e2d\uff0c\u4e8c\u500b\u9078 \\(x\\)\u00a0\u4e00\u500b\u9078 \\(y\\)\u00a0\u76f8\u4e58\u800c\u5f97\uff0c\u5982\u6b64\u9078\u53d6\u7684\u65b9\u6cd5\u6578\u70ba \\(C_1^3\\)\uff0c\u6240\u4ee5 \\(x^2y^1\\)\u00a0\u9805\u7684\u4fc2\u6578\u662f \\(C_1^3=3\\)\u3002<\/p>\n

\u4e0d\u904e\uff0c\u4e5f\u53ef\u63db\u500b\u65b9\u5f0f\u4f86\u770b \\(x^2y^1\\)\u00a0\u9805\u7684\u7522\u751f\u3002\u5982\u5716\u4e00\u6240\u793a\uff0c\u9078\u53d6\u4e8c\u500b\u9078 \\(x\\)\u3001\u4e00\u500b \\(y\\)\u00a0\u5f8c\uff0c\u5176\u60c5\u5f62\u7b49\u540c\u65bc \\(2\\) \u500b \\(x\\)\u00a0\u8207 \\(1\\) \u500b \\(y\\)\u00a0\u7684\u4e0d\u76e1\u76f8\u7570\u7269\u76f4\u7dda\u6392\u5217<\/b>\u3002\u56e0\u6b64\uff0c\\(2\\) \u500b \\(x\\)\u3001\\(1\\) \u500b \\(y\\)\u00a0\u7684\u76f4\u7dda\u6392\u5217\u53ef\u7522\u751f \\(x^2y^1\\)\u00a0\u9805\uff0c\u9019\u6a23\u7684\u6392\u5217\u65b9\u6cd5\u6578\u70ba \\(\\frac{3!}{1!2!}=3=C_1^3\\)\uff0c\u6545 \\(x^2y^1\\)\u00a0\u9805\u7684\u4fc2\u6578\u662f \\(C_1^3=3\\)\u3002<\/p>\n

\"57318_p1\"<\/a>

\u5716\u4e00\\(~~~(x+y)^3\\) \u90e8\u5206\u96c6\u9805\u793a\u610f\u5716<\/p><\/div>\n

<\/p>\n

\u9032\u4e00\u6b65\uff0c\u300c\u4e0d\u76e1\u76f8\u7570\u7269\u76f4\u7dda\u6392\u5217\u300d\u7684\u770b\u6cd5\u8b93\u6211\u5011\u53ef\u4ee5\u5c07 \\((x+y)^n\\)\u00a0\u63a8\u5ee3\u5230\u5f62\u5982 \\({({x_1} + {x_2} +\\cdots+ {x_m})^n}\\)\u00a0\u7684\u591a\u9805\u5f0f\u5b9a\u7406(Multinomial Theorem)\u3002<\/p>\n

\u9084\u662f\u7531 \\((x+y+z)^3\\)\u00a0\u8aaa\u8d77\uff0c\u540c\u6a23\u7684\uff0c\\((x+y+z)^3\\) \u662f\u00a0\\((x+y+z)\\) \u9023\u4e58\u4e09\u6b21\uff0c\u5373\u00a0\\((x+y+z)(x+y+z)(x+y+z)\\)\uff0c\u6613\u77e5\\((x+y+z)^3\\) \u7684\u5c55\u958b\u5f0f\u6703\u6709<\/p>\n

\\({x^3},{y^3},{z^3},{x^2}y,{x^2}z,{y^2}x,{y^2}z,{z^2}x,{z^2}y,xyz\\)<\/p>\n

\u5171\u8a08\u00a0\\(H_3^3 = C_3^5 = 10\\)\u00a0\u7a2e\u985e\u578b\u7684\u9f4a\u6b21\u9805\u3002<\/p>\n

\u6bcf\u9805\u4ecd\u662f\u9019\u4e09\u500b\u62ec\u865f\u4e2d\uff0c\u9078\u53d6\u4e0d\u540c\u6578\u91cf\u7684 \\(x\\)\u00a0\u6216 \\(y\\)\u00a0\u6216 \\(z\\)\u00a0\u76f8\u4e58\u800c\u5f97\uff0c\u6bcf\u9805\u7684\u4fc2\u6578\u4e00\u6a23\u5c0d\u61c9\u5230\u9078\u53d6\u7684\u65b9\u6cd5\u6578\u3002\u5982\u5716\u4e8c\u6240\u793a\uff0c\\(x^2y(=x^2y^1z^0)\\)\u00a0\u9805\u53ef\u8996\u70ba\u4e09\u500b\u62ec\u865f\u4e2d\uff0c\\(2\\) \u500b\u9078 \\(x\\)\u3001\\(1\\)\u00a0\u500b\u9078 \\(y\\)\u3001\\(0\\) \u500b\u9078 \\(z\\)\uff0c\u505a\u4e0d\u76e1\u76f8\u7570\u7269\u7684\u76f4\u7dda\u6392\u5217\uff0c\u65b9\u6cd5\u6578\u70ba \\(\\frac{3!}{2!1!0!}=3\\)\u3002\u56e0\u6b64\uff0c\\(x^2y\\)\u00a0\u9805\u7684\u4fc2\u6578\u70ba \\(3\\)\u3002<\/p>\n

\u540c\u7406\uff0c\\(xyz(=x^1y^1z^1)\\)\u00a0\u9805\u5247\u662f\u4e09\u500b\u62ec\u865f\u4e2d\uff0c\\(1\\) \u500b\u9078 \\(x\\)\u3001\\(1\\) \u500b\u9078 \\(y\\)\uff0c\\(1\\) \u500b\u9078 \\(z\\)\uff0c\u505a\u4e0d\u76e1\u76f8\u7570\u7269\u7684\u76f4\u7dda\u6392\u5217\uff0c\u65b9\u6cd5\u6578\u70ba \\(\\frac{3!}{1!1!1!}=6\\)\uff0c\u6545 \\(xyz\\)\u00a0\u7684\u4fc2\u6578\u70ba \\(6\\)\u3002<\/p>\n

\"57318_p2\"<\/a>

\u5716\u4e8c\\(~~~(x+y+z)^3\\) \u90e8\u4efd\u96c6\u9805\u793a\u610f\u5716<\/p><\/div>\n

\u56e0\u6b64\uff0c\\((x+y+z)^3\\)\u00a0\u7684\u5c55\u958b\u5f0f\u70ba<\/p>\n

\\(\\begin{array}{ll} (x + y + z)^3 &= {x^3} + {y^3} + {z^3} + \\frac{{3!}}{{2!1!0!}}{x^2}{y^1}{z^0} + \\frac{{3!}}{{2!0!1!}}{x^2}{y^0}{z^1} + \\frac{{3!}}{{1!2!0!}}{x^1}{y^2}{z^0} + \\frac{{3!}}{{0!2!1!}}{x^0}{y^2}{z^1} + \\frac{{3!}}{{1!0!2!}}{x^1}{y^0}{z^2} + \\frac{{3!}}{{0!1!2!}}{x^0}{y^1}{z^2} + \\frac{{3!}}{{1!1!1!}}{x^1}{y^1}{z^1} \\\\&=\\sum\\limits_{ a + b + c = 3 \\atop 0 \\le a,b,c \\le 3}^{}{\\frac{{3!}}{{a!b!c!}}{x^a}{y^b}{z^c}}\\end{array}\\)<\/p>\n

\u96d6\u7136\u5f0f\u5b50\u770b\u4f86\u7e41\u8907\u4e9b\uff0c\u4f46\u4fc2\u6578\u7684\u898f\u5f8b\u8207\u4e8c\u9805\u5f0f\u5b9a\u7406\u4e26\u7121\u4e8c\u81f4\u3002
\n\u53ea\u662f\u8d85\u904e\u4e8c\u9805\uff0c\u7121\u6cd5\u53ea\u7528\u7d44\u5408\u6578 \\(C\\)\u00a0\u4f86\u8868\u793a\u3002<\/p>\n

\u4eff\u7167\u4e0a\u8ff0\u770b\u6cd5\uff0c\u4e0d\u96e3\u985e\u63a8\u51fa\u5f62\u5982 \\((x_1+x_2+\\cdots+x_m)^n\\)\u00a0\u5c55\u958b\u5f0f\u4e2d\u7684\u5404\u9805\u4fc2\u6578\u8207\u5176\u5c55\u958b\u5f0f\uff0c<\/p>\n

\\(({x_1} + {x_2} +\\cdots+ {x_m})^n= \\sum\\limits_{{k_1} + {k_2} + \\cdots + {k_m} = n \\atop 0 \\le {k_1},{k_2},\\cdots,{k_m} \\le n} {\\frac{{n!}}{{{k_1}!{k_2}! \\cdots {k_m}!}}{x_1}^{{k_1}}{x_2}^{{k_2}}\\cdots{x_m}^{{k_m}}}\\)<\/p>\n

\u5176\u4e2d\u00a0\\(\\frac{{n!}}{{{k_1}!{k_2}! \\cdots {k_m}!}}{x_1}^{{k_1}}{x_2}^{{k_2}} \\cdots {x_m}^{{k_m}}\\)\u00a0\u7a31\u70ba\u5c55\u958b\u5f0f\u7684\u4e00\u822c\u9805<\/b>\u3002<\/p>\n

\u5be6\u969b\u64cd\u4f5c\u4e00\u904d\uff0c\u6bd4\u8f03\u80fd\u638c\u63e1\u7b26\u865f\u7684\u610f\u7fa9\uff0c\u8a66\u8a66 \\((x+y+z)^4\\)\u00a0\u7684\u5c55\u958b\u5f0f\uff0c
\n\u5171\u6709 \\(H_4^3=C_4^6=15\\)\u00a0\u9805<\/p>\n

\\(\\begin{array}{ll}{(x + y + z)^4} &= \\frac{{4!}}{{4!0!0!}}{x^4}{y^0}{z^0} + \\frac{{4!}}{{0!4!0!}}{x^0}{y^4}{z^0} + \\frac{{4!}}{{0!0!4!}}{x^0}{y^0}{z^4} + \\frac{{4!}}{{3!1!0!}}{x^3}{y^1}{z^0} + \\frac{{4!}}{{3!0!1!}}{x^3}{y^0}{z^1} + \\frac{{4!}}{{1!3!0!}}{x^1}{y^3}{z^0} + \\frac{{4!}}{{0!3!1!}}{x^0}{y^3}{z^1} + \\frac{{4!}}{{1!0!3!}}{x^1}{y^0}{z^3} + \\frac{{4!}}{{0!1!3!}}{x^0}{y^1}{z^3} + \\frac{{4!}}{{2!2!0!}}{x^2}{y^2}{z^0} + \\frac{{4!}}{{2!0!2!}}{x^2}{y^0}{z^2}+ \\frac{{4!}}{{0!2!2!}}{x^0}{y^2}{z^2} + \\frac{{4!}}{{2!1!1!}}{x^2}{y^1}{z^1} + \\frac{{4!}}{{1!2!1!}}{x^1}{y^2}{z^1}+\\frac{{4!}}{{1!1!2!}}{x^1}{y^1}{z^2}\\\\&= {x^4} + {y^4} + {z^4} + 4{x^3}y + 4{x^3}z + 4x{y^3} + 4{y^3}z + 4x{z^3} + 4y{z^3} + 6{x^2}{y^2} + 6{x^2}{z^2} + 6{y^2}{z^2} + 12{x^2}yz + 12x{y^2}z + 12xy{z^2}\\end{array}\\)<\/p>\n

\u7576\u5168\u90e8\u5c55\u958b\u6642\uff0c\u5f0f\u5b50\u76f8\u7576\u9f90\u96dc\uff0c\u4f46\u5176\u898f\u5f8b\u4e0d\u96e3\u638c\u63e1\uff01\u76f8\u4fe1\u518d\u8010\u5fc3\u770b\u500b\u5e7e\u904d\u61c9\u80fd\u6f38\u5165\u4f73\u5883\u3002\u6700\u597d\u7684\u6e2c\u8a66\uff0c\u5c31\u662f\u8a08\u7b97\u770b\u770b \\((x+3y-z)^6\\)\u00a0\u5c55\u958b\u5f0f\u4e2d \\(x^2yz^3\\)\u00a0\u9805\u7684\u4fc2\u6578\uff0c\u5e0c\u671b\u4f60\/\u59b3\u80fd\u5f97\u51fa\u7b54\u6848\u70ba \\(-180\\)\u3002<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u5f9e\u4e8c\u9805\u5f0f\u5b9a\u7406\u5230\u591a\u9805\u5f0f\u5b9a\u7406 (2)\uff08From Binomial Theorem to Multinomial T…<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[228,111],"tags":[3916,7163,3129],"class_list":["post-57318","post","type-post","status-publish","format-standard","hentry","category-math05","category-mathematics00","tag-3916","tag-7163","tag-3129","loop-entry","cat-228","cat-111","no-thumbnail"],"views":6608,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/57318","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=57318"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/57318\/revisions"}],"predecessor-version":[{"id":86876,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/57318\/revisions\/86876"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=57318"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=57318"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=57318"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}