{"id":6,"date":"2013-12-11T04:50:17","date_gmt":"2013-12-10T20:50:17","guid":{"rendered":"http:\/\/geld.hkto.net\/document1\/?p=6"},"modified":"2013-12-11T23:53:22","modified_gmt":"2013-12-11T15:53:22","slug":"%e7%94%a8%e7%b5%84%e5%90%88%e6%95%b8%e5%ad%b8%e4%be%86%e8%ad%89%e6%98%8e%e6%81%86%e7%ad%89%e5%bc%8f","status":"publish","type":"post","link":"http:\/\/geld.hkto.net\/document1\/?p=6","title":{"rendered":"\u7528\u7d44\u5408\u6578\u5b78\u4f86\u8b49\u660e\u6046\u7b49\u5f0f"},"content":{"rendered":"<p>\u8b49\u660e <img src='http:\/\/s0.wp.com\/latex.php?latex=%5Csum_%7Bk%3D1%7D%5E%7Bn%7Dk%5Cbinom%7Bn%7D%7Bk%7D%3Dn2%5E%7Bn-1%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\sum_{k=1}^{n}k\\binom{n}{k}=n2^{n-1}' title='\\sum_{k=1}^{n}k\\binom{n}{k}=n2^{n-1}' class='latex' \/> \uff0c\u5176\u4e2d <img src='http:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> \u662f\u6b63\u6574\u6578\u3002<\/p>\n<p>\u9019\u9053\u984c\u76ee\u662f\u4ee5\u524d\u9999\u6e2f\u9ad8\u8003\u7d14\u6578\u8a66\u5377\u4e2d\u7684\u5e38\u5ba2\u3002<\/p>\n<p><del datetime=\"2013-12-10T18:40:41+00:00\">\u8003\u8a55\u5c40\u7684\u8a55\u5206\u6a19\u6e96<\/del>\u8001\u5e2b\u6703\u6559\u6211\u5011\u8003\u616e\u4e8c\u9805\u5f0f\u5b9a\u7406\uff08Binomial Theorem\uff09\uff0c\u5f97<br \/>\n<img src='http:\/\/s0.wp.com\/latex.php?latex=%28x%2B1%29%5En%3D%5Csum_%7Bk%3D0%7D%5E%7Bn%7D%5Cbinom%7Bn%7D%7Bk%7Dx%5Ek&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='(x+1)^n=\\sum_{k=0}^{n}\\binom{n}{k}x^k' title='(x+1)^n=\\sum_{k=0}^{n}\\binom{n}{k}x^k' class='latex' \/><\/p>\n<p>\u63a5\u8457\uff0c\u628a\u65b9\u7a0b\u5f0f\u7684\u5169\u908a\u540c\u6642\u5c0d <img src='http:\/\/s0.wp.com\/latex.php?latex=x&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='x' title='x' class='latex' \/> \u5fae\u5206\uff0c\u5f97<br \/>\n<img src='http:\/\/s0.wp.com\/latex.php?latex=n%28x%2B1%29%5E%7Bn-1%7D%3D%5Csum_%7Bk%3D1%7D%5E%7Bn%7Dk%5Cbinom%7Bn%7D%7Bk%7Dx%5E%7Bk-1%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n(x+1)^{n-1}=\\sum_{k=1}^{n}k\\binom{n}{k}x^{k-1}' title='n(x+1)^{n-1}=\\sum_{k=1}^{n}k\\binom{n}{k}x^{k-1}' class='latex' \/><\/p>\n<p>\u6700\u5f8c\uff0c\u4ee3\u5165  <img src='http:\/\/s0.wp.com\/latex.php?latex=x%3D1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='x=1' title='x=1' class='latex' \/> \u5373\u5b8c\u6210\u8b49\u660e\u3002<\/p>\n<p>\u6216\u8005\u5927\u5bb6\u672a\u63a5\u89f8\u904e\u7d14\u6578\uff0c\u8b93\u6211\u5011\u8a66\u8a66\u7528\u7d44\u5408\u6578\u5b78\uff08Combinatorics\uff09\u4f86\u8b49\u660e\u3002<\/p>\n<p>\u5148\u60f3\u50cf\u4e00\u4e0b\u9019\u500b\u60c5\u666f\uff1a\u6211\u5011\u8981\u7531 <img src='http:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> \u4eba\u4e4b\u4e2d\u9078\u51fa\u4e00\u4e9b\u4eba\uff08\u6700\u5c11\u4e00\u4eba\u3001\u6700\u591a\u5168\u90e8\u4eba\uff09\u4f86\u6210\u7acb\u4e00\u500b\u59d4\u54e1\u6703\uff0c\u518d\u59d4\u6d3e\u59d4\u54e1\u6703\u5167\u7684\u5176\u4e2d\u4e00\u4eba\u7576\u6703\u9577\u3002\u59d4\u54e1\u6703\u7684\u7d44\u5408\u65b9\u6cd5\u6709\u591a\u5c11\u7a2e\u5462\uff1f<\/p>\n<p>\u6839\u64da\u5b9a\u7fa9\uff0c\u7531 <img src='http:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> \u4eba\u9078\u51fa <img src='http:\/\/s0.wp.com\/latex.php?latex=k&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='k' title='k' class='latex' \/> \u4eba\u7684\u65b9\u6cd5\u6709 <img src='http:\/\/s0.wp.com\/latex.php?latex=%5Cbinom%7Bn%7D%7Bk%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\binom{n}{k}' title='\\binom{n}{k}' class='latex' \/> \u7a2e\uff1b\u800c\u7531 k \u4eba\u4e4b\u4e2d\u9078\u51fa\u4e00\u4eba\u7576\u6703\u9577\u7684\u65b9\u6cd5\u6709 <img src='http:\/\/s0.wp.com\/latex.php?latex=k&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='k' title='k' class='latex' \/> \u7a2e\u3002\u56e0\u70ba <img src='http:\/\/s0.wp.com\/latex.php?latex=k&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='k' title='k' class='latex' \/> \u7684\u53d6\u503c\u7bc4\u570d\u662f\u7531 <img src='http:\/\/s0.wp.com\/latex.php?latex=1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='1' title='1' class='latex' \/> \u5230 <img src='http:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/>\uff0c\u6240\u4ee5\u6839\u64da\u57fa\u672c\u7684\u7d44\u5408\u6578\u5b78\u6cd5\u5247\uff0c\u7e3d\u5171\u6709 <img src='http:\/\/s0.wp.com\/latex.php?latex=%5Csum_%7Bk%3D1%7D%5E%7Bn%7Dk%5Cbinom%7Bn%7D%7Bk%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\sum_{k=1}^{n}k\\binom{n}{k}' title='\\sum_{k=1}^{n}k\\binom{n}{k}' class='latex' \/> \u7a2e\u59d4\u6d3e\u65b9\u6cd5\uff0c\u9019\u662f\u6046\u7b49\u5f0f\u7684\u5de6\u65b9\u3002<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u7531 <img src='http:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> \u4eba\u4e2d\u9078\u51fa\u4e00\u4eba\u7576\u59d4\u54e1\u6703\u7684\u6703\u9577\uff0c\u9078\u6cd5\u6709 <img src='http:\/\/s0.wp.com\/latex.php?latex=n&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n' title='n' class='latex' \/> \u7a2e\uff1b\u800c\u6703\u9577\u4ee5\u5916\u7684 <img src='http:\/\/s0.wp.com\/latex.php?latex=n-1&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n-1' title='n-1' class='latex' \/> \u4eba\u4e2d\uff0c\u6bcf\u500b\u4eba\u90fd\u6709\u5169\u7a2e\u53ef\u80fd\u6027\uff08\u6703\u54e1\u6216\u975e\u6703\u54e1\uff09\uff0c\u9078\u6cd5\u6709 <img src='http:\/\/s0.wp.com\/latex.php?latex=2%5E%7Bn-1%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='2^{n-1}' title='2^{n-1}' class='latex' \/> \u7a2e\u3002\u56e0\u6b64\uff0c\u7e3d\u5171\u6709 <img src='http:\/\/s0.wp.com\/latex.php?latex=n2%5E%7Bn-1%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='n2^{n-1}' title='n2^{n-1}' class='latex' \/> \u7a2e\u59d4\u6d3e\u65b9\u6cd5\uff0c\u9019\u662f\u6046\u7b49\u5f0f\u7684\u53f3\u65b9\u3002<\/p>\n<p>\u4ee5\u4e0a\u5169\u7a2e\u59d4\u6d3e\u65b9\u6cd5\u61c9\u8a72\u6703\u5f97\u51fa\u76f8\u7b49\u7b54\u6848\uff0c\u56e0\u6b64\u6211\u5011\u6b32\u8b49\u660e\u7684\u6046\u7b49\u5f0f\u6210\u7acb\u3002\u9019\u500b\u5c31\u662f\u7528\u7d44\u5408\u6578\u5b78\u4f86\u8b49\u660e\u6046\u7b49\u5f0f\u7684\u4f8b\u5b50\uff0c\u800c\u9019\u985e\u8b49\u660e\u5247\u88ab\u7a31\u70ba<strong>\u7d44\u5408\u8b49\u660e<\/strong>\uff08Combinatorial Proof\uff09\u3002<\/p>\n<p>\u4e00\u822c\u4f86\u8aaa\uff0c\u7d44\u5408\u8b49\u660e\u7684\u505a\u6cd5\u5c31\u662f\u5148\u865b\u69cb\u4e00\u500b\u8207\u7d44\u5408\u6578\u5b78\u6709\u95dc\u7684\u60c5\u666f\uff0c\u63a5\u8457\u8b49\u660e\u6046\u7b49\u5f0f\u7684\u5169\u908a\u90fd\u662f\uff08\u7531\u5169\u7a2e\u4e0d\u540c\u7684\u65b9\u5411\u627e\u51fa\u4f86\uff09\u9019\u500b\u554f\u984c\u7684\u7b54\u6848\u3002<\/p>\n<p>\u9084\u6709\u5f88\u591a\u6046\u7b49\u5f0f\u6709\u7d44\u5408\u8b49\u660e\uff0c\u5982 <img src='http:\/\/s0.wp.com\/latex.php?latex=%5Cbinom%7Bm%2Bn%7D%7Br%7D%3D+%5Csum_%7Bk%3D0%7D%5E%7Br%7D%5Cbinom%7Bm%7D%7Bk%7D%5Cbinom%7Bn%7D%7Br-k%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\binom{m+n}{r}= \\sum_{k=0}^{r}\\binom{m}{k}\\binom{n}{r-k}' title='\\binom{m+n}{r}= \\sum_{k=0}^{r}\\binom{m}{k}\\binom{n}{r-k}' class='latex' \/> \u548c <img src='http:\/\/s0.wp.com\/latex.php?latex=%5Cbinom%7Bn%2B1%7D%7Br%2B1%7D%3D+%5Csum_%7Bk%3Dr%7D%5E%7Bn%7D%5Cbinom%7Bk%7D%7Br%7D&#038;bg=ffffff&#038;fg=000000&#038;s=0' alt='\\binom{n+1}{r+1}= \\sum_{k=r}^{n}\\binom{k}{r}' title='\\binom{n+1}{r+1}= \\sum_{k=r}^{n}\\binom{k}{r}' class='latex' \/> \u3002<\/p>\n<p>\u7d44\u5408\u8b49\u660e\u83ef\u9e97\u548c\u751f\u52d5\u6709\u8da3\uff0c\u6bd4\u6578\u8ad6\u8b49\u660e\u53ef\u4ee5\u66f4\u6e05\u695a\u5c55\u793a\u6046\u7b49\u5f0f\u80cc\u5f8c\u7684\u610f\u7fa9\u3002\u7576\u7136\uff0c\u8981\u865b\u69cb\u4e00\u500b\u60c5\u666f\u51fa\u4f86\uff0c\u9700\u8981\u5f88\u5f37\u7684\u89c0\u5bdf\u529b\u548c\u5275\u610f\uff0c\u7d55\u5c0d\u4e0d\u662f\u4e00\u4ef6\u5bb9\u6613\u7684\u4e8b\u3002<\/p>\n<p>\u5ef6\u4f38\u95b1\u8b80\uff1a<a href=\"http:\/\/www.math.uvic.ca\/faculty\/gmacgill\/guide\/combargs.pdf\">Notes on Combinatorial Arguments from  University of Victoria<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8b49\u660e \uff0c\u5176\u4e2d \u662f\u6b63\u6574\u6578\u3002 \u9019\u9053\u984c\u76ee\u662f\u4ee5\u524d\u9999\u6e2f\u9ad8\u8003\u7d14\u6578\u8a66\u5377\u4e2d\u7684\u5e38\u5ba2\u3002 \u8003\u8a55\u5c40\u7684\u8a55\u5206\u6a19\u6e96\u8001\u5e2b\u6703\u6559\u6211\u5011\u8003\u616e\u4e8c\u9805\u5f0f\u5b9a\u7406\uff08Binomial Theorem\uff09\uff0c\u5f97 \u63a5\u8457\uff0c\u628a\u65b9\u7a0b\u5f0f\u7684\u5169\u908a\u540c\u6642\u5c0d \u5fae\u5206\uff0c\u5f97 \u6700\u5f8c\uff0c\u4ee3\u5165 \u5373\u5b8c\u6210\u8b49\u660e\u3002 \u6216\u8005\u5927\u5bb6\u672a\u63a5\u89f8\u904e\u7d14\u6578\uff0c\u8b93\u6211\u5011\u8a66\u8a66\u7528\u7d44\u5408\u6578\u5b78\uff08Combinatorics\uff09\u4f86\u8b49\u660e\u3002 \u5148\u60f3\u50cf\u4e00\u4e0b\u9019\u500b\u60c5\u666f\uff1a\u6211\u5011\u8981\u7531 \u4eba\u4e4b\u4e2d\u9078\u51fa\u4e00\u4e9b\u4eba\uff08\u6700\u5c11\u4e00\u4eba\u3001\u6700\u591a\u5168\u90e8\u4eba\uff09\u4f86\u6210\u7acb\u4e00\u500b\u59d4\u54e1\u6703\uff0c\u518d\u59d4\u6d3e\u59d4\u54e1\u6703\u5167\u7684\u5176\u4e2d\u4e00\u4eba\u7576\u6703\u9577\u3002\u59d4\u54e1\u6703\u7684\u7d44\u5408\u65b9\u6cd5\u6709\u591a\u5c11\u7a2e\u5462\uff1f \u6839\u64da\u5b9a\u7fa9\uff0c\u7531 \u4eba\u9078\u51fa \u4eba\u7684\u65b9\u6cd5\u6709 \u7a2e\uff1b\u800c\u7531 k \u4eba\u4e4b\u4e2d\u9078\u51fa\u4e00\u4eba\u7576\u6703\u9577\u7684\u65b9\u6cd5\u6709 \u7a2e\u3002\u56e0\u70ba \u7684\u53d6\u503c\u7bc4\u570d\u662f\u7531 \u5230 \uff0c\u6240\u4ee5\u6839\u64da\u57fa\u672c\u7684\u7d44\u5408\u6578\u5b78\u6cd5\u5247\uff0c\u7e3d\u5171\u6709 \u7a2e\u59d4\u6d3e\u65b9\u6cd5\uff0c\u9019\u662f\u6046\u7b49\u5f0f\u7684\u5de6\u65b9\u3002 \u53e6\u4e00\u65b9\u9762\uff0c\u7531 \u4eba\u4e2d\u9078\u51fa\u4e00\u4eba\u7576\u59d4\u54e1\u6703\u7684\u6703\u9577\uff0c\u9078\u6cd5\u6709 \u7a2e\uff1b\u800c\u6703\u9577\u4ee5\u5916\u7684 \u4eba\u4e2d\uff0c\u6bcf\u500b\u4eba\u90fd\u6709\u5169\u7a2e\u53ef\u80fd\u6027\uff08\u6703\u54e1\u6216\u975e\u6703\u54e1\uff09\uff0c\u9078\u6cd5\u6709 \u7a2e\u3002\u56e0\u6b64\uff0c\u7e3d\u5171\u6709 \u7a2e\u59d4\u6d3e\u65b9\u6cd5\uff0c\u9019\u662f\u6046\u7b49\u5f0f\u7684\u53f3\u65b9\u3002 \u4ee5\u4e0a\u5169\u7a2e\u59d4\u6d3e\u65b9\u6cd5\u61c9\u8a72\u6703\u5f97\u51fa\u76f8\u7b49\u7b54\u6848\uff0c\u56e0\u6b64\u6211\u5011\u6b32\u8b49\u660e\u7684\u6046\u7b49\u5f0f\u6210\u7acb\u3002\u9019\u500b\u5c31\u662f\u7528\u7d44\u5408\u6578\u5b78\u4f86\u8b49\u660e\u6046\u7b49\u5f0f\u7684\u4f8b\u5b50\uff0c\u800c\u9019\u985e\u8b49\u660e\u5247\u88ab\u7a31\u70ba\u7d44\u5408\u8b49\u660e\uff08Combinatorial Proof\uff09\u3002 \u4e00\u822c\u4f86\u8aaa\uff0c\u7d44\u5408\u8b49\u660e\u7684\u505a\u6cd5\u5c31\u662f\u5148\u865b\u69cb\u4e00\u500b\u8207\u7d44\u5408\u6578\u5b78\u6709\u95dc\u7684\u60c5\u666f\uff0c\u63a5\u8457\u8b49\u660e\u6046\u7b49\u5f0f\u7684\u5169\u908a\u90fd\u662f\uff08\u7531\u5169\u7a2e\u4e0d\u540c\u7684\u65b9\u5411\u627e\u51fa\u4f86\uff09\u9019\u500b\u554f\u984c\u7684\u7b54\u6848\u3002 \u9084\u6709\u5f88\u591a\u6046\u7b49\u5f0f\u6709\u7d44\u5408\u8b49\u660e\uff0c\u5982 \u548c \u3002 \u7d44\u5408\u8b49\u660e\u83ef\u9e97\u548c\u751f\u52d5\u6709\u8da3\uff0c\u6bd4\u6578\u8ad6\u8b49\u660e\u53ef\u4ee5\u66f4\u6e05\u695a\u5c55\u793a\u6046\u7b49\u5f0f\u80cc\u5f8c\u7684\u610f\u7fa9\u3002\u7576\u7136\uff0c\u8981\u865b\u69cb\u4e00\u500b\u60c5\u666f\u51fa\u4f86\uff0c\u9700\u8981\u5f88\u5f37\u7684\u89c0\u5bdf\u529b\u548c\u5275\u610f\uff0c\u7d55\u5c0d\u4e0d\u662f\u4e00\u4ef6\u5bb9\u6613\u7684\u4e8b\u3002 \u5ef6\u4f38\u95b1\u8b80\uff1aNotes on Combinatorial Arguments from University of Victoria<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"http:\/\/geld.hkto.net\/document1\/index.php?rest_route=\/wp\/v2\/posts\/6"}],"collection":[{"href":"http:\/\/geld.hkto.net\/document1\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/geld.hkto.net\/document1\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/geld.hkto.net\/document1\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/geld.hkto.net\/document1\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6"}],"version-history":[{"count":29,"href":"http:\/\/geld.hkto.net\/document1\/index.php?rest_route=\/wp\/v2\/posts\/6\/revisions"}],"predecessor-version":[{"id":37,"href":"http:\/\/geld.hkto.net\/document1\/index.php?rest_route=\/wp\/v2\/posts\/6\/revisions\/37"}],"wp:attachment":[{"href":"http:\/\/geld.hkto.net\/document1\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/geld.hkto.net\/document1\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/geld.hkto.net\/document1\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}