{"id":222,"date":"2011-12-03T00:14:39","date_gmt":"2011-12-02T15:14:39","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=222"},"modified":"2021-11-04T13:13:13","modified_gmt":"2021-11-04T04:13:13","slug":"htb200905","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb200905\/","title":{"rendered":"\u4e00\u6a4b\u59272009\uff1a\u7b2c5\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(X , Y , Z\\) \u3068\u66f8\u304b\u308c\u305f\u30ab\u30fc\u30c9\u304c\u305d\u308c\u305e\u308c \\(1\\) \u679a\u305a\u3064\u3042\u308b.\r\n\u3053\u306e\u4e2d\u304b\u3089 \\(1\\) \u679a\u306e\u30ab\u30fc\u30c9\u304c\u9078\u3070\u308c\u305f\u3068\u304d, \\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 \\(P\\) \u3092\u6b21\u306e\u898f\u5247\u306b\u3057\u305f\u304c\u3063\u3066\u79fb\u52d5\u3059\u308b.<\/p>\r\n<ul>\r\n<li><p>\\(X\\) \u306e\u30ab\u30fc\u30c9\u304c\u9078\u3070\u308c\u305f\u3068\u304d, \\(P\\) \u3092 \\(x\\) \u8ef8\u306e\u6b63\u65b9\u5411\u306b \\(1\\) \u3060\u3051\u79fb\u52d5\u3059\u308b.<\/p><\/li>\r\n<li><p>\\(Y\\) \u306e\u30ab\u30fc\u30c9\u304c\u9078\u3070\u308c\u305f\u3068\u304d, \\(P\\) \u3092 \\(y\\) \u8ef8\u306e\u6b63\u65b9\u5411\u306b \\(1\\) \u3060\u3051\u79fb\u52d5\u3059\u308b.<\/p><\/li>\r\n<li><p>\\(Z\\) \u306e\u30ab\u30fc\u30c9\u304c\u9078\u3070\u308c\u305f\u3068\u304d, \\(P\\) \u306f\u79fb\u52d5\u305b\u305a\u305d\u306e\u307e\u307e\u306e\u4f4d\u7f6e\u306b\u3068\u3069\u307e\u308b.<\/p><\/li>\r\n<\/ul>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(n\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. \u6700\u521d, \u70b9 \\(P\\) \u3092\u539f\u70b9\u306e\u4f4d\u7f6e\u306b\u304a\u304f. \\(X\\) \u306e\u30ab\u30fc\u30c9\u3068 \\(Y\\) \u306e\u30ab\u30fc\u30c9\u306e \\(2\\) \u679a\u304b\u3089\u7121\u4f5c\u70ba\u306b \\(1\\) \u679a\u3092\u9078\u3073, \\(P\\) \u3092, \u4e0a\u306e\u898f\u5247\u306b\u3057\u305f\u304c\u3063\u3066\u79fb\u52d5\u3059\u308b\u3068\u3044\u3046\u8a66\u884c\u3092 \\(n\\) \u56de\u7e70\u308a\u8fd4\u3059.\r\n<ol>\r\n<li><p><strong>(i)<\/strong>\u3000\\(n\\) \u56de\u306e\u8a66\u884c\u306e\u5f8c\u306b \\(P\\) \u304c\u5230\u9054\u53ef\u80fd\u306a\u70b9\u306e\u500b\u6570\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(ii)<\/strong>\u3000\\(P\\) \u304c\u5230\u9054\u3059\u308b\u78ba\u7387\u304c\u6700\u5927\u306e\u70b9\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol><\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(n\\) \u3092\u6b63\u306e \\(3\\) \u306e\u500d\u6570\u3068\u3059\u308b. \u6700\u521d, \u70b9 \\(P\\) \u3092\u539f\u70b9\u306e\u4f4d\u7f6e\u306b\u304a\u304f. \\(X\\) \u306e\u30ab\u30fc\u30c9, \\(Y\\) \u306e\u30ab\u30fc\u30c9, \\(Z\\) \u306e\u30ab\u30fc\u30c9\u306e \\(3\\) \u679a\u304b\u3089\u7121\u4f5c\u70ba\u306b \\(1\\) \u679a\u3092\u9078\u3073, \\(P\\) \u3092, \u4e0a\u306e\u898f\u5247\u306b\u3057\u305f\u304c\u3063\u3066\u79fb\u52d5\u3059\u308b\u3068\u3044\u3046\u8a66\u884c\u3092 \\(n\\) \u56de\u7e70\u308a\u8fd4\u3059.\r\n<ol>\r\n<li><p><strong>(i)<\/strong>\u3000\\(n\\) \u56de\u306e\u8a66\u884c\u306e\u5f8c\u306b \\(P\\) \u304c\u5230\u9054\u53ef\u80fd\u306a\u70b9\u306e\u500b\u6570\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(ii)<\/strong>\u3000\\(P\\) \u304c\u5230\u9054\u3059\u308b\u78ba\u7387\u304c\u6700\u5927\u306e\u70b9\u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol><\/p><\/li>\r\n<\/ol>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p><strong>(i)<\/strong><\/p>\r\n<p>\\(x\\) \u5ea7\u6a19\u3068 \\(y\\) \u5ea7\u6a19\u304c\u3068\u3082\u306b \\(0\\) \u4ee5\u4e0a\u3067, \u548c\u304c \\(n\\) \u3068\u306a\u308b\u70b9\u306a\u306e\u3067\r\n\\[\r\n( 0 , n ) , ( 1 , n-1 ) \\cdots ( n , 0 )\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\n\\underline{n+1} \\quad \\text{\u500b}\r\n\\]\r\n<p><strong>(ii)<\/strong><\/p>\r\n<p>\\(P\\) \u304c\u70b9 \\(( k , n-k ) \\ ( 0 \\leqq k \\leqq n )\\) \u3068\u306a\u308b\u78ba\u7387\u3092 \\(p _ k\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\np _ k & = {} _ {n} \\text{C} {} _ {k} \\left( \\dfrac{1}{2} \\right)^k \\left( \\dfrac{1}{2} \\right)^{n-k} \\\\\r\n& = \\dfrac{n!}{k! (n-k)!} \\left( \\dfrac{1}{2} \\right)^n\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(\\dfrac{p _ {k+1}}{p _ k}\\) \u3068 \\(1\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\dfrac{p _ {k+1}}{p _ k} & = \\dfrac{\\frac{n!}{(k+1)! (n-k-1)!}}{\\frac{n!}{k! (n-k)!}} \\\\\r\n& = \\dfrac{n-k}{k+1} \\gt 1 \\\\\r\n\\text{\u2234} \\quad k & \\lt \\dfrac{n-1}{2}\r\n\\end{align}\\]\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(n\\) \u304c\u5076\u6570\u306e\u3068\u304d\r\n\\[\r\n\\left\\{ \\begin{array}{ll} p _ {k+1} \\gt p _ k & \\left( \\ 1 \\leqq k \\leqq \\dfrac{n}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ p _ {k+1} \\lt p _ k & \\left( \\ \\dfrac{n}{2}+1 \\leqq k \\leqq n-1 \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\np _ 1 \\lt \\cdots \\lt p _ {\\frac{n}{2}-1} \\lt p _ {\\frac{n}{2}} \\gt p _ {\\frac{n}{2}+1} \\gt \\cdots \\gt p _ n\r\n\\]\r\n\u3086\u3048\u306b, \\(p _ {\\frac{n}{2}}\\) \u304c\u6700\u5927\u3067\u3042\u308b.<\/p><\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(n\\) \u304c\u5947\u6570\u306e\u3068\u304d\r\n\\[\r\n\\left\\{ \\begin{array}{ll} p _ {k+1} \\gt p _ k & \\left( \\ 1 \\leqq k \\leqq \\dfrac{n-3}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ p _ {k+1} = p _ k & \\left( \\ k = \\dfrac{n-1}{2} \\text{\u306e\u3068\u304d} \\right) \\\\ p _ {k+1} \\lt p _ k & \\left( \\ \\dfrac{n+1}{2} \\leqq k \\leqq n-1 \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\np _ 1 \\lt p _ 2 \\lt \\cdots \\lt p _ {\\frac{n-3}{2}} \\lt p _ {\\frac{n-1}{2}} = p _ {\\frac{n+1}{2}} \\gt p _ {\\frac{n+3}{2}} \\gt \\cdots \\gt p _ n\r\n\\]\r\n\u3086\u3048\u306b, \\(p _ {\\frac{n-1}{2}} , p _ {\\frac{n+1}{2}}\\) \u304c\u6700\u5927\u3067\u3042\u308b.<\/p><\/li>\r\n<\/ol>\r\n<p>\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u70b9\u306f<\/p>\r\n<ul>\r\n<li><p>\\(n\\) \u304c\u5076\u6570\u306e\u3068\u304d, \\(\\underline{\\left( \\dfrac{n}{2} , \\dfrac{n}{2} \\right)}\\)<\/p><\/li>\r\n<li><p>\\(n\\) \u304c\u5947\u6570\u306e\u3068\u304d, \\(\\underline{\\left( \\dfrac{n-1}{2} , \\dfrac{n+1}{2} \\right) , \\left( \\dfrac{n+1}{2} , \\dfrac{n-1}{2} \\right)}\\)<\/p><\/li>\r\n<\/ul>\r\n<p><strong>(2)<\/strong><\/p>\r\n<p><strong>(i)<\/strong><\/p>\r\n<p>\\(X\\) \u304c \\(n-k-\\ell\\) \u56de, \\(Y\\) \u304c \\(k\\) \u56de, \\(Z\\) \u304c \\(\\ell\\) \u56de\uff08 \\(0 \\leqq k+\\ell \\leqq n\\) \uff09\u9078\u3070\u308c\u305f\u3068\u304d, \u70b9 \\(P\\) \u306f \\(( n-k-\\ell , k )\\) \u306b\u79fb\u52d5\u3059\u308b.<br \/>\r\n\\(0 \\leqq \\ell \\leqq n\\) \u306a\u306e\u3067, <strong>(1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070, \u6c42\u3081\u308b\u500b\u6570\u306f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {\\ell=0}^{n} (n-\\ell+1) & = \\textstyle\\sum\\limits _ {\\ell=0}^{n} (\\ell+1) \\\\\r\n& = \\underline{\\dfrac{1}{2} (n+1)(n+2)} \\quad \\text{\u500b}\r\n\\end{align}\\]\r\n<p><strong>(ii)<\/strong><\/p>\r\n<p>\\(\\ell\\) \u3092\u5b9a\u6570\u3068\u307f\u306a\u305b\u3070, <strong>(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a<\/p>\r\n<ul>\r\n<li><p>\\(n-\\ell\\) \u304c\u5076\u6570\u306e\u3068\u304d, \u70b9 \\(\\left( \\dfrac{n-\\ell}{2} , \\dfrac{n-\\ell}{2} \\right)\\) \u306b\u9054\u3059\u308b\u78ba\u7387\r\n\\[\r\n\\dfrac{n!}{\\left( \\frac{n-\\ell}{2} \\right)! \\left( \\frac{n-\\ell}{2} \\right)! \\ell !} \\left( \\dfrac{1}{3} \\right)^n\r\n\\]<\/li>\r\n<li><p>\\(n-\\ell\\) \u304c\u5947\u6570\u306e\u3068\u304d, \u70b9 \\(\\left( \\dfrac{n-\\ell pm 1}{2} , \\dfrac{n-\\ell \\mp 1}{2} \\right)\\) \u306b\u9054\u3059\u308b\u78ba\u7387\r\n\\[\r\n\\dfrac{n!}{\\left( \\frac{n-\\ell-1}{2} \\right)! \\left( \\frac{n-\\ell+1}{2} \\right)! \\ell !} \\left( \\dfrac{1}{3} \\right)^n\r\n\\]<\/li>\r\n<\/ul>\r\n<p>\u304c\u6700\u5927\u3068\u306a\u308b. \u3053\u306e\u78ba\u7387\u3092 \\(q _ {\\ell}\\) \u3068\u304a\u304f.<\/p>\r\n<ol>\r\n<li><p><strong>1*<\/strong>\u3000\\(n-\\ell\\) \u304c\u5076\u6570\u306e\u3068\u304d<br \/>\r\n\\(\\dfrac{q _ {\\ell+1}}{q _ {\\ell}}\\) \u3068 \\(1\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\dfrac{q _ {\\ell+1}}{q _ {\\ell}} & = \\dfrac{\\dfrac{n!}{\\left( \\frac{n-\\ell}{2} \\right) ! \\left( \\frac{n-\\ell}{2}-1 \\right) ! ( \\ell+1 ) !}}{\\dfrac{n!}{\\left( \\frac{n-\\ell}{2} \\right) ! \\left( \\frac{n-\\ell}{2} \\right)! \\ell !}} \\\\\r\n& = \\dfrac{n-\\ell}{2(\\ell+1)} \\gt 1 \\\\\r\n\\text{\u2234} \\quad \\ell & \\lt \\dfrac{n-2}{3}\r\n\\end{align}\\]\r\n\\(n\\) \u306f \\(3\\) \u306e\u500d\u6570\u306a\u306e\u3067\r\n\\[\r\nq _ 0 \\lt \\cdots \\lt q _ {\\frac{n}{3}-1} \\lt q _ {\\frac{n}{3}} \\gt q _ {\\frac{n}{3}+1} \\gt \\cdots \\gt q _ n\r\n\\]<\/li>\r\n<li><p><strong>2*<\/strong>\u3000\\(n-\\ell\\) \u304c\u5947\u6570\u306e\u3068\u304d<br \/>\r\n\\(\\dfrac{q _ {\\ell+1}}{q _ {\\ell}}\\) \u3068 \\(1\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\dfrac{q _ {\\ell+1}}{q _ {\\ell}} & = \\dfrac{\\dfrac{n!}{\\left( \\frac{n-\\ell-1}{2} \\right) ! \\left( \\frac{n-\\ell+1}{2}-1 \\right) ! ( \\ell+1 ) !}}{\\dfrac{n!}{\\left( \\frac{n-\\ell-1}{2} \\right) ! \\left( \\frac{n-\\ell+1}{2} \\right)! \\ell !}} \\\\\r\n& = \\dfrac{n-\\ell+1}{2(\\ell+1)} \\gt 1 \\\\\r\n\\text{\u2234} \\quad \\ell & \\lt \\dfrac{n-1}{3}\r\n\\end{align}\\]\r\n\\(n\\) \u306f \\(3\\) \u306e\u500d\u6570\u306a\u306e\u3067\r\n\\[\r\nq _ 0 \\lt \\cdots \\lt q _ {\\frac{n}{3}-1} \\lt q _ {\\frac{n}{3}} \\gt q _ {\\frac{n}{3}+1} \\gt \\cdots \\gt q _ n\r\n\\]<\/li>\r\n<\/ol>\r\n<p><strong>1*<\/strong> <strong>2*<\/strong> \u3088\u308a, \\(\\ell = \\dfrac{n}{3}\\) \u306e\u3068\u304d, \\(q _ {\\ell}\\) \u306f\u6700\u5927\u3068\u306a\u308b.<br \/>\r\n\u3053\u306e\u3068\u304d\r\n\\[\r\nn-\\ell = n-\\dfrac{n}{3} = \\dfrac{2n}{3}\r\n\\]\r\n\u3067\u5076\u6570\u3068\u306a\u308b\u3053\u3068\u304b\u3089, \u6c42\u3081\u308b\u70b9\u306f\r\n\\[\r\n\\underline{\\left( \\dfrac{n}{3} , \\dfrac{n}{3} \\right)}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(X , Y , Z\\) \u3068\u66f8\u304b\u308c\u305f\u30ab\u30fc\u30c9\u304c\u305d\u308c\u305e\u308c \\(1\\) \u679a\u305a\u3064\u3042\u308b. \u3053\u306e\u4e2d\u304b\u3089 \\(1\\) \u679a\u306e\u30ab\u30fc\u30c9\u304c\u9078\u3070\u308c\u305f\u3068\u304d, \\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 \\(P\\) \u3092\u6b21\u306e\u898f\u5247\u306b\u3057\u305f\u304c\u3063\u3066\u79fb\u52d5\u3059\u308b. \\(X\\) \u306e &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/htb200905\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[45],"tags":[146,15],"class_list":["post-222","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2009","tag-hitotsubashi","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/222","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=222"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/222\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=222"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=222"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=222"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}