{"id":1262,"date":"2015-09-03T22:54:54","date_gmt":"2015-09-03T13:54:54","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1262"},"modified":"2021-09-13T19:12:28","modified_gmt":"2021-09-13T10:12:28","slug":"iks201501","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks201501\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272015\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(n\\) \u3092\u81ea\u7136\u6570, \\(m\\) \u3092 \\(2n\\) \u4ee5\u4e0b\u306e\u81ea\u7136\u6570\u3068\u3059\u308b.\r\n\\(1\\) \u304b\u3089 \\(n\\) \u307e\u3067\u306e\u81ea\u7136\u6570\u304c \\(1\\) \u3064\u305a\u3064\u8a18\u3055\u308c\u305f\u30ab\u30fc\u30c9\u304c, \u305d\u308c\u305e\u308c\u306e\u6570\u306b\u5bfe\u3057\u3066 \\(2\\) \u679a\u305a\u3064, \u5408\u8a08 \\(2n\\) \u679a\u3042\u308b.\r\n\u3053\u306e\u4e2d\u304b\u3089, \\(m\\) \u679a\u306e\u30ab\u30fc\u30c9\u3092\u7121\u4f5c\u70ba\u306b\u9078\u3093\u3060\u3068\u304d, \u305d\u308c\u3089\u306b\u8a18\u3055\u308c\u305f\u6570\u304c\u3059\u3079\u3066\u7570\u306a\u308b\u78ba\u7387\u3092 \\(P _ n (m)\\) \u3068\u8868\u3059.\r\n\u305f\u3060\u3057 \\(P _ n (1) = 1\\) \u3068\u3059\u308b. \u3055\u3089\u306b, \\(E _ n (m) = m P _ n (m)\\) \u3068\u304a\u304f. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(P _ 3 (2) , P _ 3 (3) , P _ 3 (4)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(E _ {10} (m)\\) \u3092\u6700\u5927\u306b\u3059\u308b\u3088\u3046\u306a \\(m\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057, \\(E _ n (m) \\gt E _ n (m+1)\\) \u3092\u6e80\u305f\u3059\u81ea\u7136\u6570 \\(m\\) \u306e\u6700\u5c0f\u5024\u3092 \\(f(n)\\) \u3068\u3059\u308b\u3068\u304d, \\(f(n)\\) \u3092 \\(n\\) \u3092\u7528\u3044\u3066\u8868\u305b.\r\n\u305f\u3060\u3057, \u30ac\u30a6\u30b9\u8a18\u53f7 \\([ \\quad ]\\) \u3092\u7528\u3044\u3066\u3088\u3044. \u3053\u3053\u3067, \u5b9f\u6570 \\(x\\) \u306b\u5bfe\u3057\u3066, \\(x\\) \u3092\u8d85\u3048\u306a\u3044\u6700\u5927\u306e\u6574\u6570\u3092 \\([x]\\) \u3068\u8868\u3059.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \u30ab\u30fc\u30c9\u306e\u53d6\u308a\u51fa\u3057\u65b9\u306f, \\({} _ {2n} \\text{C} {} _ {m}\\) \u901a\u308a\u3042\u308b.
    \r\n\u6761\u4ef6\u3092\u307f\u305f\u3059\u30ab\u30fc\u30c9\u306e\u53d6\u308a\u51fa\u3057\u65b9\u306f \\(1 \\leqq m \\leqq n\\) \u306e\u3068\u304d\u306b\u9650\u3063\u3066, \\(2^m {} _ {n} \\text{C} {} _ {m}\\) \u901a\u308a\u3042\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nP _ n (m) & = \\left\\{ \\begin{array}{ll} \\dfrac{2^m {} _ {n} \\text{C} {} _ {m}}{{} _ {2n} \\text{C} {} _ {m}} & ( \\ 1 \\leqq m \\leqq n \\ \\text{\u306e\u3068\u304d} ) \\\\ 0 & ( \\ n+1 \\leqq m \\leqq 2n \\ \\text{\u306e\u3068\u304d} ) \\end{array} \\right. \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nP _ 3 (2) & = \\dfrac{2^2 {} _ {3} \\text{C} {} _ {2}}{{} _ {6} \\text{C} {} _ {2}} = \\dfrac{4 \\cdot 3}{15} = \\underline{\\dfrac{4}{5}} , \\\\\r\nP _ 3 (3) & = \\dfrac{2^3 {} _ {3} \\text{C} {} _ {3}}{{} _ {6} \\text{C} {} _ {3}} = \\dfrac{8}{20} = \\underline{\\dfrac{2}{5}} , \\\\\r\nP _ 3 (4) & = \\underline{0} \\ .\r\n\\end{align}\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\(1 \\leqq m \\leqq n\\) \u306e\u5834\u5408\u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u3044.
    \r\n\u3053\u306e\u3068\u304d, \\(1 \\leqq m \\leqq n-1\\) \u306b\u5bfe\u3057\u3066\r\n\\[\\begin{align}\r\n\\dfrac{E _ n (m+1)}{E _ n (m)} & = \\dfrac{(m+1) 2^{m+1} {} _ {n} \\text{C} {} _ {m+1}}{{} _ {2n} \\text{C} {} _ {m+1}} \\cdot \\dfrac{{} _ {2n} \\text{C} {} _ {m}}{m 2^m {} _ {n} \\text{C} {} _ {m}} \\\\\r\n& = \\dfrac{2 (m+1)}{m} \\cdot \\dfrac{n !}{(m+1) ! (n-m-1) !} \\cdot \\dfrac{m ! (n-m) !}{n !} \\\\\r\n& \\qquad \\cdot \\dfrac{2n !}{m ! (2n-m) !} \\cdot \\dfrac{(m+1) ! (2n-m-1) !}{2n !} \\\\\r\n& = \\dfrac{2 (m+1) (n-m)}{m (2n-m)} \\quad ... [1] \\ .\r\n\\end{align}\\]\r\n\\(n = 10\\) \u306e\u3068\u304d, [1] \u3068 \\(1\\) \u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n\\dfrac{2 (m+1) (10-m)}{m (20-m)} & \\gt 1 \\\\\r\n2 ( 10 +9m -m^2 ) & \\gt 20m -m^2 \\quad ( \\ \\text{\u2235} \\ m (20-m) \\gt 0 \\ ) \\\\\r\nm^2 +2m -20 & \\lt 0 \\\\\r\n\\text{\u2234} \\quad -1 -\\sqrt{21} & \\lt m \\lt -1 +\\sqrt{21} \\ .\r\n\\end{align}\\]\r\n\\(4 \\lt \\sqrt{21} \\lt 5\\) \u306a\u306e\u3067\r\n\\[\r\n1 \\leqq m \\leqq 3 \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\left\\{ \\begin{array}{ll} E _ {10} (m+1) \\gt E _ {10} (m) & ( \\ 1 \\leqq m \\leqq 3 \\ \\text{\u306e\u3068\u304d} ) \\\\ E _ {10} (m+1) \\lt E _ {10} (m) & ( \\ 4 \\leqq m \\leqq 9 \\ \\text{\u306e\u3068\u304d} ) \\end{array} \\right. \\ .\r\n\\]\r\n\u306a\u306e\u3067, \u6c42\u3081\u308b \\(m\\) \u306e\u5024\u306f\r\n\\[\r\nm = \\underline{4} \\ .\r\n\\]\r\n

    (3)<\/strong><\/p>\r\n

    (2)<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308b\u3068\r\n\\[\\begin{align}\r\n\\dfrac{2 (m+1) (n-m)}{m (2n-m)} & \\gt 1 \\\\\r\n2 \\{ n +(n-1) m -m^2 \\} & \\gt 2nm -m^2 \\quad ( \\ \\text{\u2235} \\ m (20-m) \\gt 0 \\ ) \\\\\r\nm^2 +2m -2n & \\lt 0 \\\\\r\n\\text{\u2234} \\quad -1 -\\sqrt{2n+1} & \\lt m \\lt -1 +\\sqrt{2n+1} \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n1 \\leqq m \\leqq \\left[ \\sqrt{2n+1} \\right] -1 \\ .\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(m\\) \u306e\u5024\u306f\r\n\\[\r\nm = \\underline{\\left[ \\sqrt{2n+1} \\right]} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092\u81ea\u7136\u6570, \\(m\\) \u3092 \\(2n\\) \u4ee5\u4e0b\u306e\u81ea\u7136\u6570\u3068\u3059\u308b. \\(1\\) \u304b\u3089 \\(n\\) \u307e\u3067\u306e\u81ea\u7136\u6570\u304c \\(1\\) \u3064\u305a\u3064\u8a18\u3055\u308c\u305f\u30ab\u30fc\u30c9\u304c, \u305d\u308c\u305e\u308c\u306e\u6570\u306b\u5bfe\u3057\u3066 \\(2\\) \u679a\u305a\u3064, \u5408\u8a08 \\(2n\\ … \u7d9a\u304d\u3092\u8aad\u3080 →<\/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":[130],"tags":[145,137],"class_list":["post-1262","post","type-post","status-publish","format-standard","hentry","category-ikashika_2015","tag-ikashika","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1262","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=1262"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1262\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1262"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1262"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1262"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}