{"id":13424,"date":"2024-03-19T18:02:31","date_gmt":"2024-03-19T18:02:31","guid":{"rendered":"https:\/\/science-hub.click\/%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E4%BA%8C%E9%A0%85%E5%85%AC%E5%BC%8F%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC\/"},"modified":"2024-03-19T18:02:31","modified_gmt":"2024-03-19T18:02:31","slug":"%E3%83%8B%E3%83%A5%E3%83%BC%E3%83%88%E3%83%B3%E4%BA%8C%E9%A0%85%E5%85%AC%E5%BC%8F%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=13424","title":{"rendered":"\u30cb\u30e5\u30fc\u30c8\u30f3\u4e8c\u9805\u516c\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac"},"content":{"rendered":"

\u30cb\u30e5\u30fc\u30c8\u30f3\u306e\u4e8c\u9805\u5f0f<\/strong>\u306f\u3001\u4e8c\u9805\u5f0f\u306e\u6574\u6570\u3079\u304d\u4e57\u306e\u5c55\u958b\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u30a2\u30a4\u30b6\u30c3\u30af \u30cb\u30e5\u30fc\u30c8\u30f3\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u305f\u6570\u5f0f\u3067\u3059\u3002\u3053\u308c\u306f\u3001\u30cb\u30e5\u30fc\u30c8\u30f3\u306e\u4e8c\u9805\u516c\u5f0f<\/strong>\u3001\u307e\u305f\u306f\u3088\u308a\u5358\u7d14\u306b\u4e8c\u9805<\/span>\u516c\u5f0f<\/strong>\u3068\u3082\u547c\u3070\u308c\u307e\u3059\u3002<\/p>

\u58f0\u660e<\/span><\/h2>

\u30ea\u30f3\u30b0\u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u9805x<\/i>\u3068y<\/i> ( xy=yx<\/i>\u306a\u3069) \u3068\u81ea\u7136\u6570<\/a><\/span>n<\/i>\u3067\u69cb\u6210\u3055\u308c\u308b\u4e8c\u9805\u5f0f\u3092\u8003\u3048\u307e\u3059\u3002 <\/p>

$$ {(x+y)^n=\\sum_{k=0}^n {n \\choose k} x^{n-k} y^k} $$<\/div><\/dd><\/dl>

\u3053\u3053\u3067\u6570\u5b57\u306f<\/p>

$$ {{n \\choose k}=\\frac{n!}{k!\\,(n-k)!}} $$<\/div><\/dd><\/dl>

(\u6642\u3005\u6ce8\u610f\u3055\u308c\u308b\u3053\u3068\u3082\u3042\u308a\u307e\u3059)

$$ {C_n^k} $$<\/div> ) \u306f\u4e8c\u9805\u4fc2\u6570\u3067\u3059\u3002<\/p>

\u5f0f\u306ey \u3092<\/i>-y<\/i>\u306b\u7f6e\u304d\u63db\u3048\u308b\u3068\u30012 \u756a\u76ee\u306e\u9805\u304c\u8ca0\u306e\u5024\u3068\u3057\u3066\u6271\u308f\u308c\u307e\u3059\u3002 <\/p>

$$ {(x-y)^n=\\Big(x+(-y)\\Big)^n=\\sum_{k=0}^n {n \\choose k} x^{n-k} (-y)^k} $$<\/div><\/dd><\/dl>

\u4f8b \uff1a <\/p>

$$ {n=2~,\\qquad(x + y)^2 = x^2 + 2xy + y^2\\,} $$<\/div><\/dd>
$$ {n=3~,\\qquad(x – y)^3 = x^3 – 3x^2y + 3xy^2 – y^3\\,} $$<\/div><\/dd>
$$ {n=4~,\\qquad(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\\,} $$<\/div><\/dd><\/dl>
\n\"\u30cb\u30e5\u30fc\u30c8\u30f3\u4e8c\u9805\u516c\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u30c7\u30e2\u30f3\u30b9\u30c8\u30ec\u30fc\u30b7\u30e7\u30f3<\/a><\/span><\/span><\/h2>

x<\/i> \u3001 y \u3092<\/i>xy=yx<\/i>\u3068\u306a\u308b\u3088\u3046\u306a\u74b0\u306e\u8981\u7d20\u3068\u3057\u3001 n \u3092<\/i>\u81ea\u7136\u6570\u3068\u3057\u307e\u3059\u3002 <\/p>

$$ {(x+y)^n=\\sum_{k=0}^n {n \\choose k} x^{n-k} y^k} $$<\/div><\/dd><\/dl>

\u3053\u306e\u516c\u5f0f\u3092\u5e30\u7d0d\u6cd5\u3067\u8a3c\u660e\u3057\u307e\u3057\u3087\u3046\u3002<\/p>

\n\"\u30cb\u30e5\u30fc\u30c8\u30f3\u4e8c\u9805\u516c\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u521d\u671f\u5316<\/span><\/h3>
$$ {n=0~,\\qquad(x+y)^0=1={0 \\choose 0}x^0y^0} $$<\/div><\/dd>
$$ {n=1~,\\qquad(x+y)^1= x + y ={1 \\choose 0}x^1y^0 + {1 \\choose 1}x^0y^1} $$<\/div><\/dd><\/dl>
\n\"\u30cb\u30e5\u30fc\u30c8\u30f3\u4e8c\u9805\u516c\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u907a\u4f1d<\/a><\/span><\/span><\/h3>

n \u3092<\/i>1 \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3057\u3001\u3053\u306e\u95a2\u4fc2\u304cn<\/i>\u306b\u771f\u3067\u3042\u308b\u5834\u5408\u3001 n+1<\/i>\u306b\u3082\u771f\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u307e\u3059\u3002<\/p>

\u518d\u767a\u4eee\u8aac\u306b\u3088\u308a\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 <\/p>

$$ {(x+y)^{n+1}=(x+y)\\cdot\\sum_{k=0}^n {n \\choose k} x^{n-k} y^k,} $$<\/div><\/dd><\/dl>

\u306e\u5206\u914d\u6027<\/a><\/span>\u306b\u3088\u308a\u3001

$$ {\\cdot} $$<\/div> +<\/span>\u3067: <\/p>
$$ {(x+y)^{n+1}=x^{n+1}+x\\cdot\\sum_{k=1}^n {n \\choose k} x^{n-k} y^k +y\\cdot\\sum_{k=0}^{n-1} {n \\choose k} x^{n-k} y^k + y^{n+1}} $$<\/div><\/dd><\/dl>

\u56e0\u6570\u5206\u89e3\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 <\/p>

$$ {(x+y)^{n+1} =x^{n+1}+\\sum_{k=1}^n \\left\\lbrack {{n} \\choose {k}} + {{n} \\choose {k-1}} \\right\\rbrack x^{n-k+1} y^{k}+ y^{n+1}} $$<\/div><\/dd><\/dl>

\u30d1\u30b9\u30ab\u30eb\u306e\u4e09\u89d2<\/a><\/span>\u516c\u5f0f\u3092\u4f7f\u7528\u3059\u308b\u3068\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 <\/p>

$$ {(x+y)^{n+1} =x^{n+1}+\\sum_{k=1}^n {{n+1}\\choose k}~x^{n-k+1} y^{k}+y^{n+1}} $$<\/div><\/dd><\/dl>

\u3053\u308c\u3067\u30c7\u30e2\u30f3\u30b9\u30c8\u30ec\u30fc\u30b7\u30e7\u30f3\u306f\u7d42\u4e86\u3067\u3059\u3002<\/p><\/div>

\n\"\u30cb\u30e5\u30fc\u30c8\u30f3\u4e8c\u9805\u516c\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. Binomiaalstelling \u2013 afrikaans<\/a><\/li>
  2. \u0645\u0628\u0631\u0647\u0646\u0629 \u0630\u0627\u062a \u0627\u0644\u062d\u062f\u064a\u0646 \u2013 arabe<\/a><\/li>
  3. \u0411\u0456\u043d\u043e\u043c \u041d\u044c\u044e\u0442\u0430\u043d\u0430 \u2013 bi\u00e9lorusse<\/a><\/li>
  4. \u041d\u044e\u0442\u043e\u043d\u043e\u0432 \u0431\u0438\u043d\u043e\u043c \u2013 bulgare<\/a><\/li>
  5. Teorema binomial \u2013 banjar<\/a><\/li>
  6. \u09a6\u09cd\u09ac\u09bf\u09aa\u09a6\u09c0 \u0989\u09aa\u09aa\u09be\u09a6\u09cd\u09af \u2013 bengali<\/a><\/li><\/ol><\/div>\n
    \n

    \n \u30cb\u30e5\u30fc\u30c8\u30f3\u4e8c\u9805\u516c\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n