{"id":312,"date":"2012-02-23T00:42:23","date_gmt":"2012-02-22T15:42:23","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=312"},"modified":"2021-10-30T15:24:06","modified_gmt":"2021-10-30T06:24:06","slug":"wsr200804","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr200804\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52008\uff1a\u7b2c4\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(n\\) \u500b\u306e\u7403\u3068 \\(n\\) \u500b\u306e\u7bb1\u304c\u3042\u308b.\r\n\u5404\u7403\u3092\u7121\u4f5c\u70ba\u306b\u3069\u308c\u304b\u306e\u7bb1\u306b\u5165\u308c\u308b. \u3059\u306a\u308f\u3061\u5404\u7403\u3092\u72ec\u7acb\u306b\u78ba\u7387 \\(\\dfrac{1}{n}\\) \u3067\u3069\u308c\u304b \\(1\\) \u3064\u306e\u7bb1\u306b\u5165\u308c\u308b\u3082\u306e\u3068\u3059\u308b. \\(n \\geqq 3\\) \u306e\u3068\u304d, \\(2\\) \u7bb1\u306e\u307f\u304c\u7a7a\u306b\u306a\u308b\u78ba\u7387\u3092 \\(p _ n\\) \u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(p _ 3 , p _ 4\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(n \\geqq 4\\) \u3068\u3059\u308b. \\(2\\) \u7bb1\u306e\u307f\u304c\u7a7a\u3067, \\(1\\) \u7bb1\u306b \\(3\\) \u500b\u306e\u7403\u304c\u5165\u308a, \u305d\u306e\u4ed6\u306e \\((n-3)\\) \u500b\u306e\u305d\u308c\u305e\u308c\u306b \\(1\\) \u500b\u306e\u7403\u304c\u5165\u308b\u78ba\u7387 \\(q _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\\(n \\geqq 5\\) \u306b\u5bfe\u3057, \\(p _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(4)<\/strong>\u3000<strong>(2)<\/strong> \u3067\u6c42\u3081\u305f \\(q _ n\\) \u306b\u3064\u3044\u3066, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{q _ n}{p _ n}\\) \u3092\u6c42\u3081\u3088.<\/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<ul>\r\n<li><p>\\(n=3\\) \u306e\u5834\u5408<br \/>\r\n\u7403\u306e\u5165\u308c\u65b9\u306f\u5168\u90e8\u3067 \\(3^3\\) \u901a\u308a\u3042\u308b.<br \/>\r\n\\(2\\) \u7bb1\u304c\u7a7a\u306b\u306a\u308b\u306e\u306f, \\(3\\) \u3064\u306e\u7403\u304c\u3059\u3079\u3066 \\(1\\) \u7bb1\u306b\u5165\u308b\u3068\u304d\u3067, \u3053\u306e\u65b9\u6cd5\u306f \\({} _ {3} \\text{C} {} _ {3} =3\\) \u901a\u308a\u3042\u308b.<br \/>\r\n\u3088\u3063\u3066\r\n\\[\r\np _ 3 = \\dfrac{3}{3^3} =\\underline{\\dfrac{1}{9}}\r\n\\]<\/li>\r\n<li><p>\\(n=4\\) \u306e\u5834\u5408<br \/>\r\n\u7403\u306e\u5165\u308c\u65b9\u306f\u5168\u90e8\u3067 \\(4^4\\) \u901a\u308a\u3042\u308b.<br \/>\r\n\\(2\\) \u7bb1\u304c\u7a7a\u306b\u306a\u308b\u306e\u306f, \\(2\\) \u7bb1\u306b \\(4\\) \u3064\u306e\u7403\u304c\u5206\u304b\u308c\u3066\u5165\u308b\u3068\u304d\u3067, \u3053\u306e\u65b9\u6cd5\u306f, \u7bb1\u306e\u9078\u3073\u65b9\u304c \\({} _ {4} \\text{C} {} _ {2} =6\\) \u901a\u308a, \u7403\u306e\u5206\u3051\u65b9\u304c \\(2^4 -2 =14\\) \u901a\u308a\u3042\u308b\uff08\u3059\u3079\u3066\u306e\u7403\u304c \\(1\\) \u7bb1\u306b\u5165\u308b\u5834\u5408\u3092\u9664\u3044\u305f\uff09.<br \/>\r\n\u3088\u3063\u3066\r\n\\[\r\np _ 4 = \\dfrac{6 \\cdot 14}{4^4} = \\underline{\\dfrac{21}{64}}\r\n\\]<\/li>\r\n<\/ul>\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\u7403\u306e\u5165\u308c\u65b9\u306f\u5168\u90e8\u3067 \\(n^n\\) \u901a\u308a\u3042\u308b.<br \/>\r\n\\(n\\) \u7bb1\u304b\u3089, \u7a7a\u306b\u306a\u308b \\(2\\) \u7bb1\u3068 \\(3\\) \u500b\u306e\u7403\u304c\u5165\u308b \\(1\\) \u7bb1\u306e\u9078\u3073\u65b9\u306f\r\n\\[\r\n{} _ {n} \\text{C} {} _ {2} \\cdot {} _ {n-2} \\text{C} {} _ {1} = \\dfrac{n(n-1)(n-2)}{2} \\quad \\text{\u901a\u308a}\r\n\\]\r\n\\(n\\) \u500b\u306e\u7403\u3092, \u540c\u3058\u7bb1\u306b \\(3\\) \u500b, \u6b8b\u308a \\((n-3)\\) \u500b\u3092 \\((n-3)\\) \u7bb1\u306b \\(1\\) \u500b\u305a\u3064\u5165\u308c\u308b\u65b9\u6cd5\u306f\r\n\\[\r\n{} _ {n} \\text{C} {} _ {3} \\cdot (n-3) ! = \\dfrac{n!}{6} \\quad \\text{\u901a\u308a}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nq _ n = \\dfrac{\\dfrac{n(n-1)(n-2)}{2} \\cdot \\dfrac{n!}{6}}{n^n} = \\underline{\\dfrac{(n-1)(n-2) n!}{12 n^{n-1}}}\r\n\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<p>\\(2\\) \u7bb1\u304c\u7a7a\u306b\u306a\u308a, \\(2\\) \u7bb1\u306b \\(2\\) \u500b\u305a\u3064, \\((n-4)\\) \u7bb1\u306b \\(1\\) \u3064\u305a\u3064\u7403\u304c\u5165\u308b\u78ba\u7387\u3092 \\(r _ n\\) \u3068\u304a\u304f.<br \/>\r\n\\(n\\) \u500b\u306e\u7bb1\u306e\u3046\u3061, \u7a7a\u306b\u306a\u308b \\(2\\) \u7bb1, \u7403\u304c \\(2\\) \u500b\u5165\u308b \\(2\\) \u7bb1\u306e\u9078\u3073\u65b9\u306f\r\n\\[\r\n{} _ {n} \\text{C} {} _ {2} \\cdot {} _ {n-2} \\text{C} {} _ {2} = \\dfrac{n(n-1)(n-2)(n-3)}{4} \\quad \\text{\u901a\u308a}\r\n\\]\r\n\\(n\\) \u500b\u306e\u7403\u3092, \\(2\\) \u7bb1\u306b \\(2\\) \u500b\u305a\u3064, \u6b8b\u308a \\((n-4)\\) \u500b\u3092 \\((n-4)\\) \u7bb1\u306b \\(1\\) \u500b\u305a\u3064\u5165\u308c\u308b\u65b9\u6cd5\u306f\r\n\\[\r\n{} _ {n} \\text{C} {} _ {2} \\cdot {} _ {n-2} \\text{C} {} _ {2} \\cdot (n-4) ! = \\dfrac{n!}{4} \\quad \\text{\u901a\u308a}\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\nr _ n & = \\dfrac{\\dfrac{n(n-1)(n-2)(n-3)}{4} \\cdot \\dfrac{n!}{4}}{n^n} \\\\\r\n& = \\dfrac{(n-1)(n-2)(n-3) n!}{16 n^{n-1}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, <strong>(2)<\/strong> \u306e\u7d50\u679c\u3082\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\np _ n & = q _ n+r _ n \\\\\r\n& = \\dfrac{(n-1)(n-2) \\left\\{ 4+3(n-3) \\right\\} n!}{48 n^{n-1}} \\\\\r\n& = \\underline{\\dfrac{(n-1)(n-2)(3n-5) n!}{48 n^{n-1}}}\r\n\\end{align}\\]\r\n<p><strong>(4)<\/strong><\/p>\r\n<p><strong>(2)<\/strong> , <strong>(3)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{q _ n}{p _ n} = \\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{4}{3n-5} = \\underline{0}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u500b\u306e\u7403\u3068 \\(n\\) \u500b\u306e\u7bb1\u304c\u3042\u308b. \u5404\u7403\u3092\u7121\u4f5c\u70ba\u306b\u3069\u308c\u304b\u306e\u7bb1\u306b\u5165\u308c\u308b. \u3059\u306a\u308f\u3061\u5404\u7403\u3092\u72ec\u7acb\u306b\u78ba\u7387 \\(\\dfrac{1}{n}\\) \u3067\u3069\u308c\u304b \\(1\\) \u3064\u306e\u7bb1\u306b\u5165\u308c\u308b\u3082\u306e\u3068\u3059\u308b. \\(n \\geqq 3\\ &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/wsr200804\/\">\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":[54],"tags":[147,16],"class_list":["post-312","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2008","tag-waseda_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/312","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=312"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/312\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=312"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=312"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=312"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}