{"id":57319,"date":"2014-09-19T01:02:41","date_gmt":"2014-09-18T17:02:41","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=57319"},"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%861%ef%bc%88from-binomial-theorem-to-multinomial-theorem-1%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%861%ef%bc%88from-binomial-theorem-to-multinomial-theorem-1%ef%bc%89\/","title":{"rendered":"\u5f9e\u4e8c\u9805\u5f0f\u5b9a\u7406\u5230\u591a\u9805\u5f0f\u5b9a\u7406 (1)"},"content":{"rendered":"

\u5f9e\u4e8c\u9805\u5f0f\u5b9a\u7406\u5230\u591a\u9805\u5f0f\u5b9a\u7406 (1)\uff08From Binomial Theorem to Multinomial Theorem (1)\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

\u570b\u4e2d\u6642\u5b78\u5230\u4e58\u6cd5\u516c\u5f0f \\({(x + y)^2} = {x^2} + 2xy + {y^2}\\)\uff0c\\({(x + y)^3} = {x^3} + 3{x^2}y + 3x{y^2} + {y^3}\\)\uff0c\u5c31\u5728\u731c\u60f3\u00a0\\({(x + y)^4},{(x + y)^5},\\cdots,{(x + y)^n}\\)\u00a0\u5c55\u958b\u5f8c\u7684\u6a21\u6a23\u3002\u900f\u904e\u6bd4\u5c0d\u53ef\u770b\u51fa \\((x+y)^n\\)\u00a0\u7684\u5404\u9805\u90fd\u662f\u9f4a\u6b21\uff0c\u4e5f\u5c31\u662f\u8aaa\uff0c\u5c55\u958b\u7684\u5404\u9805 \\(x^ay^b\\)\u00a0\u90fd\u6703\u6eff\u8db3 \\(a+b=n\\)\u3002<\/p>\n

\u56e0\u6b64\uff0c\u53ea\u8981\u80fd\u638c\u63e1\u5404\u9805\u4fc2\u6578\u7684\u898f\u5247\uff0c\u4efb\u610f\u7684\u81ea\u7136\u6578 \\(n\\)\uff0c\u6211\u5011\u4fbf\u80fd\u5c07 \\((x+y)^n\\)\u00a0\u7684\u5404\u9805\u4f9d \\(x\\)\u00a0\u6216 \\(y\\)\u00a0\u7684\u5347\u51aa\u6216\u964d\u51aa\u6392\u51fa\u3002\u570b\u4e2d\u8001\u5e2b\u63a1\u7528\u7684\u65b9\u6cd5\u662f\u5c07\u5df4\u65af\u5361\u4e09\u89d2\u5f62\u756b\u51fa(\u5716\u4e00)\uff0c\u4e00\u4e00\u5c0d\u61c9\uff0c\u53ea\u8981\u8db3\u5920\u8010\u5fc3\uff0c\u5c31\u80fd\u9054\u5230\u4efb\u610f\u7684\u81ea\u7136\u6578 \\(n\\)\u3002<\/p>\n

\"57319_p1\"<\/a>

\u5716\u4e00\\(~~~\\)\u5df4\u65af\u5361\u4e09\u89d2\u5f62<\/p><\/div>\n

<\/p>\n

\u9019\u500b\u95dc\u65bc \\((x+y)^n\\)\u00a0\u5c55\u958b\u5f0f\u6bcf\u9805\u4fc2\u6578\u7684\u898f\u5247\uff0c\u5c31\u662f\u521d\u7b49\u4ee3\u6578\u4e2d\u91cd\u8981\u7684\u4e8c\u9805\u5f0f\u5b9a\u7406\uff08Binomial Theorem\uff09\u3002\u4e8b\u5be6\u4e0a\uff0c\u900f\u904e\u6392\u5217\u7d44\u5408\u7684\u76f8\u95dc\u77e5\u8b58\uff0c\u5c31\u80fd\u63a8\u8ad6\u51fa \\((x+y)^n\\)\u00a0\u5c55\u958b\u5f0f\u5404\u9805\u7684\u4fc2\u6578\u898f\u5247\u3002\u9019\u4e5f\u70ba\u4f55\u9ad8\u4e2d\u6578\u5b78\u5c07\u4e8c\u9805\u5f0f\u5b9a\u7406\u7576\u6210\u6392\u5217\u7d44\u5408\u7684\u4e00\u500b\u61c9\u7528\u3002
\n\u9996\u5148\uff0c\u5f9e \\((x+y)^3\\)\u00a0\u7684\u5c55\u958b\u904e\u7a0b\u89c0\u5bdf\u8d77\u3002\u7531\u65bc<\/p>\n

\\(\\begin{array}{ll}{\\left( {x + y} \\right)^3} &= \\left( {x + y} \\right)\\left( {x + y} \\right)\\left( {x + y} \\right) \\\\&= xxx + xxy + xyx + yxx + xyy + yxy + yyx + yyy \\\\&= {x^3} + 3{x^2}y + 3x{y^2} + {y^3}\\end{array}\\)<\/p>\n

\u63db\u8a00\u4e4b\uff0c\\((x+y)^3\\)\u00a0\u662f \\((x+y)\\)\u00a0\u9023\u4e58\u4e09\u6b21\uff0c\u5373 \\((x+y)(x+y)(x+y)\\)\u3002\u5c55\u958b\u5f97\u5230\u7684\u5404\u9805\u76f8\u7576\u65bc\u5728\u6bcf\u4e00\u500b\u62ec\u865f\u4e2d\uff0c\u5404\u9078\u4e00\u500b \\(x\\)\u00a0\u6216 \\(y\\)\u00a0\u4f5c\u4e58\u7a4d\u800c\u5f97\u3002\u7531\u65bc\u4e58\u6cd5\u53ef\u4ee5\u4ea4\u63db\uff0c\u9032\u4e00\u6b65\u80fd\u6574\u7406\u6210 \\(x^3,x^2y^1,x^1y^2,y^3\\)\u00a0\u56db\u7a2e\u985e\u578b\u7684\u9805\uff0c\u800c\u540c\u985e\u9805\u7684\u500b\u6578\u5c31\u662f\u6bcf\u9805\u7684\u4fc2\u6578\u3002<\/p>\n

\u4e5f\u5c31\u662f\u8aaa\uff0c\u5f9e\u9019\u4e09\u500b\u62ec\u865f\u4e2d\uff0c\u500b\u5225\u9078\u53d6 \\(x\\)\u00a0\u6216 \\(y\\)\u00a0\u4f86\u4f5c\u4e58\u7a4d\uff0c\u5176\u65b9\u6cd5\u6578\u5c31\u662f\u6bcf\u500b\u5c0d\u61c9\u9805\u7684\u4fc2\u6578\u3002\u8209\u4f8b\u4f86\u8aaa\uff0c\u5982\u5716\u4e8c\u6240\u793a\uff0c\\(x^2y^1\\)\u00a0\u662f\u4e09\u500b\u62ec\u865f\u4e2d\uff0c\\(2\\) \u500b\u9078 \\(x\\)\uff0c\\(1\\) \u500b\u9078 \\(y\\)\uff0c\u5b83\u7684\u65b9\u6cd5\u6709 \\(C^3_1=3\\)\u00a0\u7a2e\uff0c\u6545\u9805\u4fc2\u6578\u53ef\u4ee5\u7d44\u5408\u6578 \\(C_1^3\\)\u00a0\u8868\u793a\u3002<\/p>\n

\"57319_p2\"<\/a>

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

\u56e0\u6b64\uff0c\u6211\u5011\u53ef\u4f9d\u9078\u53d6 \\(y\\)\u00a0\u7684\u500b\u6578\u4e4b\u898f\u5f8b\uff0c\u5beb\u51fa \\((x+y)^3\\)\u00a0\u7684\u5c55\u958b\u5f0f (\u8acb\u6ce8\u610f\u5f0f\u4e2d\u7d05\u5b57\u7684\u5c0d\u61c9\u95dc\u4fc2)<\/p>\n

\"57319_eq1\"<\/a><\/p>\n

\u4eff\u6b64\u4f5c\u6cd5\uff0c\u6211\u5011\u80fd\u985e\u63a8\u51fa \\((x+y)^n\\)\u00a0\u7684\u5c55\u958b\u5f0f\uff0c\u540c\u6a23\u8acb\u8b80\u8005\u6ce8\u610f\u5f0f\u4e2d\u7d05\u5b57\u7684\u5c0d\u61c9\u95dc\u4fc2\uff1a<\/p>\n

\"57319_eq2\"<\/a><\/p>\n

\u56e0\u6b64\uff0c\u4e8c\u9805\u5f0f\u5b9a\u7406\u70ba<\/p>\n

\u5c0d\u65bc\u4efb\u610f\u6b63\u6574\u6578 \\(n\\)\uff0c\u6211\u5011\u6709\u4e0b\u5217\u7684\u4e8c\u9805\u5c55\u958b\u5f0f\uff1a<\/p>\n

\\(\\begin{array}{ll}{\\left( {x + y} \\right)^n} &= C{\\kern 1pt} _0^n{x^n} + C_1^n{x^{n – 1}}{y^1} +\\cdots+ C{\\kern 1pt} _r^n{x^{n – r}}{y^r} +\\cdots+ C{\\kern 1pt} _{n – 1}^n{x^1}{y^{n – 1}} + C{\\kern 1pt} _n^n{y^n}{\\kern 1pt} {\\kern 1pt} {\\kern 1pt} {\\kern 1pt} {\\kern 1pt}\\\\&= \\sum\\limits_{r = 0}^n {C{\\kern 1pt} _r^n{x^{n – r}}{y^r}}\\end{array}\\)<\/p>\n

\u00a0\u5176\u4e2d\uff0c\u5f62\u5982 \\(C_r^nx^{n-r}y^r\\)\u00a0\u88ab\u7a31\u70ba\u5c55\u958b\u5f0f\u7684\u4e00\u822c\u9805<\/b>\u3002<\/p>\n

\u82e5\u5c07\u5df4\u65af\u5361\u4e09\u89d2\u5f62\u91cd\u65b0\u7528\u7d44\u5408\u6578\u5beb\u4e0a\u4e00\u904d(\u5982\u5716\u4e09)\uff0c\u6613\u77e5\u5df4\u65af\u5361\u4e09\u89d2\u5f62\u7684\u69cb\u9020\u898f\u5247\u70ba \\(C_m^n = C_{m – 1}^{n – 1} + C_m^{n – 1}\\)\uff0c\\(1\\le m\\le n-1\\)\u3002\u9019\u6b63\u662f\u9019\u500b\u7d44\u5408\u6046\u7b49\u5f0f\u88ab\u7a31\u70ba\u5df4\u65af\u5361\u5b9a\u7406<\/b>\u7684\u539f\u56e0\u3002\u4e8b\u5be6\u4e0a\uff0c\u6211\u5011\u4e5f\u80fd\u7528\u6578\u5b78\u6b78\u7d0d\u6cd5\u548c\u5df4\u65af\u5361\u5b9a\u7406\u8b49\u660e\u4e8c\u9805\u5f0f\u5b9a\u7406\uff0c\u6709\u8208\u8da3\u7684\u8b80\u8005\u4e0d\u59a8\u5617\u8a66\u770b\u770b\u3002<\/p>\n

\"57319_p3\"<\/a>

\u5716\u4e09\\(~~~\\)\u5df4\u65af\u5361\u4e09\u89d2\u5f62\u4e2d\u5404\u9805\u8207\u7d44\u5408\u6578\u7684\u5c0d\u61c9<\/p><\/div>\n

\u6709\u4e86\u4e8c\u9805\u5f0f\u5b9a\u7406\uff0c\u5c0d\u65bc\u8655\u7406\u9ad8\u6b21\u5c55\u958b\u5e6b\u52a9\u751a\u5927\uff0c\u6bd4\u65b9\u8aaa\uff0c\u591a\u9805\u5f0f\u51fd\u6578 \\(f(x)=(1+x)^{10}\\)\u00a0\u7684\u5c55\u958b\u5f0f<\/p>\n

\\(f(x) = {(1 + x)^{10}} = C_0^{10} + C_1^{10}x + C_2^{10}{x^2} +\\cdots+ C_{10}^{10}{x^{10}}\\)<\/p>\n

\u4e5f\u4f7f\u5f97\u5f62\u5982 \\((1.01)^{10}\\)\u00a0\u7684\u4f30\u8a08\uff0c\u9664\u4e86\u5229\u7528\u5c0d\u6578\u53d6\u503c\u67e5\u8868\uff0c\u53c8\u591a\u4e86\u4e00\u7a2e\u65b9\u6cd5\uff0c\u7531\u65bc<\/p>\n

\\(\\begin{array}{ll}{(1.01)^{10}} &= f(0.01) \\\\&= C_0^{10} + C_1^{10}(0.01) + C_2^{10}{(0.01)^2} +\\cdots+ C_{10}^{10}{(0.01)^{10}}\\end{array}\\)<\/p>\n

\u53d6\u524d\u56db\u9805\u5373\u53ef\u6c42\u503c\u6e96\u78ba\u5230\u5c0f\u6578\u9ede\u4ee5\u4e0b\u7b2c\u4e09\u4f4d\uff0c<\/p>\n

\\(C_0^{10} + C_1^{10}(0.01) + C_2^{10}{(0.01)^2} + C_3^{10}{(0.01)^3} \\\\= 1 + 10 \\times (0.01) + 45 \\times {(0.01)^2} + 120 \\times {(0.01)^3} \\\\= 1 + 0.1 + 0.0045 + 0.00012 = 1.10462 \\approx 1.1046\\)<\/p>\n

\u4e0d\u904e\uff0c\u82e5\u662f\u8b8a\u6578\u8d85\u904e\u4e8c\u9805\uff0c\u8a72\u5982\u4f55\u8655\u7406\u5462\uff1f\u4f8b\u5982\uff0c\\((x+y+z)^3\\)\u00a0\u7684\u5c55\u958b\u5f0f\uff0c\u6211\u5011\u53ef\u4ee5\u9019\u9ebc\u505a<\/p>\n

\\(\\begin{array}{ll}{(x + y + z)^3} &= {[(x + y) + z]^3} \\\\&= C_0^3{(x + y)^3} + C_1^3{(x + y)^2}z + C_2^3(x + y){z^2} + C_3^3{z^3} \\\\&= C_0^3(C_0^3{x^3} + C_1^3{x^2}y + C_2^3x{y^2} + C_3^3{y^3}) + C_1^3(C_0^2{x^2} + C_1^2xy + C_2^2{y^2})z + C_2^3(x + y){z^2} + C_3^3{z^3} \\\\&= {x^3} + {y^3} + {z^3} + 3{x^2}y + 3{x^2}z + 3x{y^2} + 3{y^2}z + 3x{z^2} + 3y{z^2} + 6xyz\\end{array}\\)<\/p>\n

\u5f9e\u904e\u7a0b\u4f86\u770b\uff0c\u53ea\u8981\u53cd\u8986\u904b\u7528\u4e8c\u9805\u5f0f\u5b9a\u7406\uff0c\u9084\u662f\u80fd\u8655\u7406\u4e8c\u9805\u4ee5\u4e0a\u7684\u5c55\u958b\u5f0f\u3002\u4f46\u8b93\u4eba\u597d\u5947\u7684\u662f\uff0c\u662f\u5426\u6709\u985e\u4f3c\u4e8c\u9805\u5c55\u958b\u7684\u5b9a\u7406\u5462\uff1f\u6709\u7684\uff0c\u7b54\u6848\u662f\u591a\u9805\u5f0f\u5b9a\u7406(Multinomial Theorem)\u3002\u66f4\u8b93\u4eba\u8208\u596e\u7684\u662f\uff0c\u63a8\u8ad6\u5b83\u7684\u65b9\u6cd5\u4e5f\u548c\u672c\u6587\u6240\u8ac7\u7684\u4e8c\u9805\u5f0f\u5b9a\u7406\u76f8\u540c\u3002\u6709\u8208\u8da3\u7684\u8b80\u8005\u4e0d\u59a8\u5617\u8a66\u770b\u770b\uff0c\u6211\u5011\u5c07\u5728\u3008\u5f9e\u4e8c\u9805\u5f0f\u5b9a\u7406\u5230\u591a\u9805\u5f0f\u5b9a\u7406(2)\u3009\u4e2d\u63ed\u66c9\u3002<\/p>\n

\u9023\u7d50\uff1a\u5f9e\u4e8c\u9805\u5f0f\u5b9a\u7406\u5230\u591a\u9805\u5f0f\u5b9a\u7406(2)<\/a><\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u5f9e\u4e8c\u9805\u5f0f\u5b9a\u7406\u5230\u591a\u9805\u5f0f\u5b9a\u7406 (1)\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":[3916,7163,7162,3129],"class_list":["post-57319","post","type-post","status-publish","format-standard","hentry","category-math05","category-mathematics00","tag-3916","tag-7163","tag-7162","tag-3129","loop-entry","cat-228","cat-111","no-thumbnail"],"views":21269,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/57319","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=57319"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/57319\/revisions"}],"predecessor-version":[{"id":86877,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/57319\/revisions\/86877"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=57319"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=57319"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=57319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}