{"id":3260,"date":"2025-09-23T22:11:07","date_gmt":"2025-09-23T13:11:07","guid":{"rendered":"https:\/\/math-friend.com\/?p=3260"},"modified":"2025-09-23T22:23:01","modified_gmt":"2025-09-23T13:23:01","slug":"%e3%80%90%e6%9d%b1%e4%ba%ac%e5%a4%a7%e5%ad%a6%e5%85%a5%e8%a9%a6%e3%80%91%e4%ba%8c%e9%a0%85%e4%bf%82%e6%95%b0%e3%81%ae%e6%9c%80%e5%a4%a7%e5%85%ac%e7%b4%84%e6%95%b0%e3%81%a8%e5%b8%b0%e7%b4%8d%e6%b3%952","status":"publish","type":"post","link":"https:\/\/math-friend.com\/?p=3260","title":{"rendered":"\u3010\u6771\u4eac\u5927\u5b66\u5165\u8a66\u3011\u4e8c\u9805\u4fc2\u6570\u306e\u6700\u5927\u516c\u7d04\u6570\u3068\u5e30\u7d0d\u6cd5(2009)"},"content":{"rendered":"\n<p>\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059.<\/p>\n\n\n\n<p class=\"has-border -border02\">\\(2\\)\u4ee5\u4e0a\u306e\u81ea\u7136\u6570\\(m\\)\u306b\u5bfe\u3057\u3066, \\(m-1\\)\u500b\u306e\u4e8c\u9805\u4fc2\u6570$$<br>{}_m \\mathrm{C}_1,\\,\\,{}_m \\mathrm{C}_2,\\,\\,{}_m \\mathrm{C}_3,\\,\\,\\cdots,\\,\\,{}_m \\mathrm{C}{m-1}<br>$$<br>\u3092\u8003\u3048, \u3053\u308c\u3089\u3059\u3079\u3066\u306e\u6700\u5927\u516c\u7d04\u6570\u3092\\(d_m\\)\u3068\u3059\u308b.<br>(1) \\(m\\)\u304c\u7d20\u6570\u306e\u3068\u304d, \\(d_m=m\\)\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b.<br>(2) \u3059\u3079\u3066\u306e\u81ea\u7136\u6570\\(k\\)\u306b\u5bfe\u3057\u3066, \\(k^m-k\\)\u304c\\(d_m\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u3092, \\(k\\)\u306b\u95a2\u3059\u308b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u305b.<br><span style=\"text-align:right;display:block;\">(2009 \u6771\u4eac\u5927\u5b66 \u6587\u7cfb [2])<\/span><\/p>\n\n\n\n<p>\u305d\u308c\u3067\u306f\u89e3\u3044\u3066\u3044\u304d\u307e\u3057\u3087\u3046.<\/p>\n\n\n\n<div class=\"wp-block-group is-style-crease\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<p>(1) \\({}_m \\mathrm{C}_1=m\\)\u3067, \\(m\\)\u306f\u7d20\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089, \\(d_m\\)\u306f\\(1\\)\u307e\u305f\u306f\\(m\\)\u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b. <br><br>\u6b21\u306b, \\(1\\leq k\\leq m-1\\)\u3092\u6e80\u305f\u3059\u81ea\u7136\u6570\\(k\\)\u306b\u95a2\u3057\u3066,$$<br>{}_m \\mathrm{C}_k=\\frac{m!}{k!(m-k)!}=\\frac{m\\cdot(m-1)!}{k!(m-k)!}<br>$$\u3067\u3042\u308a, \\(m&gt;k\\), \\(m&gt;m-k\\)\u3067\u3042\u308b\u304b\u3089, \u3053\u306e\u5206\u6bcd\u306f\\(m\\)\u672a\u6e80\u306e\u81ea\u7136\u6570\u306e\u7a4d\u3068\u306a\u3063\u3066\u3044\u308b. \u3088\u3063\u3066, \u5206\u6bcd\u306e\u7d20\u56e0\u6570\u306f\u3044\u305a\u308c\u3082\\(m\\)\u672a\u6e80\u3068\u306a\u308a, \u5206\u5b50\u306e\u7d20\u6570\\(m\\)\u304c\u7d04\u5206\u3055\u308c\u308b\u3053\u3068\u306f\u306a\u304f, \\({}_m \\mathrm{C}_k\\)\u306f\\(m\\)\u306e\u500d\u6570\u3068\u306a\u308b.<br><br>\u3053\u308c\u304b\u3089, \\(m\\)\u306f\\({}_m \\mathrm{C}_1,\\cdots,{}_m \\mathrm{C}{m-1}\\)\u306e\u516c\u7d04\u6570\u3068\u306a\u3063\u3066\u304a\u308a, \u6700\u5927\u516c\u7d04\u6570\\(d_m\\)\u306f\\(d_m\\geq m\\)\u3092\u6e80\u305f\u3059. \u5148\u306e\u6761\u4ef6\u3067, \\(d_m=1\\), \u307e\u305f\u306f\\(d_m=m\\)\u3067\u3042\u3063\u305f\u304b\u3089, \\(d_m=m\\)\u304c\u308f\u304b\u308b.<\/p>\n\n\n\n<p>(2) \u3059\u3079\u3066\u306e\u81ea\u7136\u6570\\(k\\)\u306b\u5bfe\u3057\u3066, \\(k^m-k\\)\u304c\\(d_m\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\\(k\\)\u306b\u95a2\u3059\u308b\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3067\u8a3c\u660e\u3059\u308b.<br>\u2460 \\(k=1\\)\u306e\u3068\u304d, \\(k^m-k=1^m-1=0\\)\u3068\u306a\u308a, \u3053\u308c\u306f\\(d_m\\)\u3067\u5272\u308a\u5207\u308c\u308b\u304b\u3089\u6210\u308a\u7acb\u3064.<br><br>\u2461 \\(k=l\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u3068\u4eee\u5b9a\u3057\u3066, \\(k=l+1\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u3059.<br>\\(k=l\\)\u306e\u3068\u304d\u6210\u308a\u7acb\u3064\u306e\u3067, \\(l^m-l\\)\u306f\\(d_m\\)\u3067\u5272\u308a\u5207\u308c\u308b\u304b\u3089, \u6574\u6570\\(A\\)\u304c\u5b58\u5728\u3057\u3066, \\(l^m-l=d_m\\times A\\)\u3068\u8868\u305b\u308b.<br><br>\\(k=l+1\\)\u306e\u3068\u304d, $$<br>\\begin{align}<br>(l+1)^m-(l+1)&amp;=\\sum_{n=0}^m {}_m \\mathrm{C}_n l^n-l-1\\\\[1.5ex]<br>&amp;=l^m+1+\\sum_{n=1}^{m-1} {}_m \\mathrm{C}_n l^n-l-1\\\\[1.5ex]<br>&amp;=d_m\\times A+\\sum_{n=1}^{m-1} {}_m \\mathrm{C}_n l^n\\\\[1.5ex]<br>\\end{align}<br>$$\u3068\u306a\u308b. \u3053\u3053\u3067, \\(1\\geq n\\geq m-1\\)\u306e\u3068\u304d, \\({}_m \\mathrm{C}_n\\)\u306f\\(d_m\\)\u3067\u5272\u308a\u5207\u308c\u308b\u306e\u3067, \u81ea\u7136\u6570\\(a_n\\)\u304c\u5b58\u5728\u3057\u3066\\({}_m \\mathrm{C}_n=d_m\\times a_n\\)\u3068\u304b\u3051\u308b\u304b\u3089, $$<br>\\begin{align}<br>(l+1)^m-(l+1)&amp;=d_m\\times A+\\sum_{n=1}^{m-1} d_m\\times a_n l^n\\\\[1.5ex]<br>&amp;=d_m\\times A+d_m\\sum_{n=1}^{m-1}  a_n l^n\\\\[1.5ex]<br>&amp;=d_m\\left(A+\\sum_{n=1}^{m-1}  a_n l^n\\right)<br>\\end{align}<br>$$\u3068\u306a\u308a, \\((l+1)^m-(l+1)\\)\u304c\\(d_m\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u304c\u308f\u304b\u308b. \u3088\u3063\u3066, \\(k=l+1\\)\u306e\u3068\u304d\u3082\u6210\u308a\u7acb\u3064.<br><br>\u2460, \u2461\u304b\u3089\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306b\u3088\u308a, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570\\(k\\)\u306b\u5bfe\u3057\u3066, \\(k^m-k\\)\u304c\\(d_m\\)\u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f.<\/p>\n<\/div><\/div>\n\n\n\n<p>youtube\u3067\u3082\u89e3\u8aac\u3057\u3066\u3044\u307e\u3059.<\/p>\n\n\n\n<iframe width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/6vtcn9R5eEE?si=n2uCOGuPhDzVzU_9\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n","protected":false},"excerpt":{"rendered":"<p>\u4eca\u56de\u306f\u3053\u3061\u3089\u306e\u554f\u984c\u3092\u89e3\u3044\u3066\u3044\u304d\u307e\u3059. \\(2\\)\u4ee5\u4e0a\u306e\u81ea\u7136\u6570\\(m\\)\u306b\u5bfe\u3057\u3066, \\(m-1\\)\u500b\u306e\u4e8c\u9805\u4fc2\u6570$${}_m \\mathrm{C}_1,\\,\\,{}_m \\mathrm{C}_2,\\,\\,{}_m \\math [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":3262,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"swell_btn_cv_data":"","_themeisle_gutenberg_block_has_review":false,"footnotes":""},"categories":[6],"tags":[],"class_list":["post-3260","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-tokimakuru"],"_links":{"self":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/3260","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3260"}],"version-history":[{"count":8,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/3260\/revisions"}],"predecessor-version":[{"id":3270,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/posts\/3260\/revisions\/3270"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=\/wp\/v2\/media\/3262"}],"wp:attachment":[{"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3260"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3260"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-friend.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3260"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}