{"id":1098,"date":"2025-12-24T17:20:36","date_gmt":"2025-12-24T08:20:36","guid":{"rendered":"https:\/\/math-travel.com\/?p=1098"},"modified":"2026-02-11T17:02:44","modified_gmt":"2026-02-11T08:02:44","slug":"factorial","status":"publish","type":"post","link":"https:\/\/math-travel.jp\/math-a\/factorial\/","title":{"rendered":"\u968e\u4e57(\uff01)\u3068\u306f\uff1f\u516c\u5f0f\u3068\u8a08\u7b97\u65b9\u6cd5\u30010\u306e\u968e\u4e57\u304c1\u306b\u306a\u308b\u7406\u7531\u3082\u5206\u304b\u308a\u3084\u3059\u304f\u89e3\u8aac"},"content":{"rendered":"\n

\u300c\u968e\u4e57\u3063\u3066\u306a\u3093\u3060\u3063\u3051\uff1f\u300d<\/span>
\u300c0\u306e\u968e\u4e57\u306f\uff1f\u5206\u6570\u306e\u968e\u4e57\u306f\uff1f\u300d<\/span><\/p>\n\n\n\n

\u4eca\u56de\u306f\u5834\u5408\u306e\u6570\u306b\u95a2\u3059\u308b\u60a9\u307f\u3092\u89e3\u6c7a\u3057\u307e\u3059\u3002<\/p>\n\n\n

\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u968e\u4e57\u306e\u8a08\u7b97\u3067\u56f0\u3063\u3066\u3066\u2026<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u5834\u5408\u306e\u6570\u3092\u6c42\u3081\u3066\u3044\u304f\u306a\u304b\u3067\u3001\u3068\u3064\u305c\u3093\u968e\u4e57\uff08\u30d3\u30c3\u30af\u30ea\u30de\u30fc\u30af\u300c\uff01\u300d\uff09\u304c\u51fa\u3066\u304d\u3066\u304a\u3069\u308d\u3044\u305f\u65b9\u3082\u591a\u3044\u306f\u305a\u3002<\/p>\n\n\n\n

\u3057\u304b\u3057\u3001\u968e\u4e57\u306e\u8a08\u7b97\u306f\u96e3\u3057\u3044\u3082\u306e\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n\n\n\n

\u3057\u3063\u304b\u308a\u3068\u968e\u4e57\u306e\u610f\u5473\u3068\u8a08\u7b97\u306e\u4ed5\u65b9\u3092\u7406\u89e3\u3059\u308c\u3070\u3001\u591a\u304f\u306e\u554f\u984c\u306f\u30b9\u30e9\u30b9\u30e9\u3068\u89e3\u3051\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u672c\u8a18\u4e8b\u3067\u306f\u3001\u968e\u4e57\u306e\u610f\u5473\u3068\u3055\u307e\u3056\u307e\u306a\u8a08\u7b97\u65b9\u6cd5\u3092\u8a73\u3057\u304f\u89e3\u8aac<\/span>\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u968e\u4e57\u3068\u306f\uff1f<\/h2>\n\n\n\n

\\(n\\)\u306e\u968e\u4e57\u3068\u306f\u300c1\u304b\u3089\\(n\\)\u307e\u3067\u306e\u3059\u3079\u3066\u306e\u81ea\u7136\u6570\u306e\u7a4d\u300d<\/span>\u3092\u6307\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u305f\u3068\u3048\u3070\u30013\u306e\u968e\u4e57\u306f\\(3 \\times 2 \\times 1=6\\)\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u968e\u4e57\u306e\u8a08\u7b97
\\[n!=n \\cdot (n-1) \\cdot (n-2) \\cdots 2 \\cdot 1\\]<\/div>\n\n\n\n

\u307e\u305f\u3001\u968e\u4e57\u3092\u8868\u3059\u3068\u304d\u306f\u6570\u5b57\u3084\u6587\u5b57\u306e\u3046\u3057\u308d\u306b\u300c\uff01\u300d\uff08\u30a8\u30af\u30b9\u30af\u30e9\u30e1\u30fc\u30b7\u30e7\u30f3\u30de\u30fc\u30af\uff09\u3092\u3064\u3051\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\u3064\u307e\u308a\u3001\\(n\\)\u306e\u968e\u4e57\u3092\\(n!\\)\u3068\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\n

\\[4!=4 \\cdot 3 \\cdot 2 \\cdot 1=24\\]<\/p>\n\n\n\n

\\[5!=5 \\cdot 4 \\cdot 3 \\cdot 2 \\cdot 1=120\\]<\/p>\n<\/div><\/div>\n\n\n\n

\u3053\u306e\u3088\u3046\u306b\u3001\u968e\u4e57\u306f\u305d\u306e\u6570\u304b\u30891\u307e\u3067\u306e\u7a4d\u3092\u6307\u3057\u3066\u3044\u308b\u306e\u3067\u3059\u3002<\/p>\n\n\n

\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u306a\u3093\u3060\u3001\u610f\u5916\u3068\u304b\u3093\u305f\u3093\u3067\u3059\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n

\"\"\u30b7\u30fc\u30bf<\/span><\/div>
\n

\u3068\u3064\u305c\u3093\u30d3\u30c3\u30af\u30ea\u30de\u30fc\u30af\u304c\u51fa\u3066\u304d\u305f\u3089\u3073\u3063\u304f\u308a\u3059\u308b\u3088\u306d<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u968e\u4e57\u306f\u9806\u5217\u306e\u8003\u3048\u65b9<\/h2>\n\n\n\n

\u968e\u4e57\u306f\u4eba\u3084\u30e2\u30ce\u306e\u4e26\u3079\u65b9\u3092\u8003\u3048\u308b\u3068\u304d\u306b\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u305f\u3068\u3048\u3070\u30011~5\u307e\u3067\u306e\u6570\u5b57\u304c\u66f8\u304b\u308c\u305f\u30ab\u30fc\u30c9\u30921\u679a\u305a\u3064\u7528\u610f\u3057\u3066\u30015\u679a\u30921\u5217\u306b\u4e26\u3079\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\"<\/figure>\n<\/div>\n\n\n

\u3053\u306e\u3068\u304d\u30ab\u30fc\u30c9\u306e\u4e26\u3079\u65b9\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8a08\u7b97\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\\[5 \\times 4 \\times 3 \\times 2\\times 1=120\\]<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u30015\u679a\u306e\u30ab\u30fc\u30c9\u3092\u4e26\u3079\u308b\u3068\u304d\u306e\u4e26\u3079\u65b9\u306f\\(5!\\)\u3068\u306a\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u307e\u305f\u30015\u679a\u306e\u30ab\u30fc\u30c9\u306e\u4e26\u3079\u65b9\u306f\u9806\u5217\u3067\u3082\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

5\u679a\u306e\u30ab\u30fc\u30c9\u304b\u30895\u679a\u3092\u53d6\u308a\u51fa\u3057\u3066\u4e26\u3079\u308b\u306e\u3067\u3001<\/p>\n\n\n\n

\\[_{5}P_{5}=5 \\times 4 \\times 3 \\times 2\\times 1=120\\]<\/p>\n\n\n\n

\u540c\u3058\u304f120\u901a\u308a\u3068\u3044\u3046\u7b54\u3048\u3092\u51fa\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

\u3064\u307e\u308a\u3001n!\u3068\u3044\u3046\u306e\u306fn\u500b\u306e\u7570\u306a\u308b\u3082\u306e\u304b\u3089n\u500b\u3092\u53d6\u308a\u51fa\u3057\u3066\u4e26\u3079\u308b\u9806\u5217\u3067\u3042\u308b\u3053\u3068\u304c\u5206\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

\\[n!=_{n}P_{n}=n \\cdot (n-1) \\cdot (n-2) \\cdots 3 \\cdot 2 \\cdot1\\]<\/div>\n\n\n
\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u968e\u4e57\u306f\u4e26\u3079\u65b9\u306e\u7dcf\u6570\u3092\u6c42\u3081\u308b\u3068\u304d\u306b\u4f7f\u3048\u308b\u3093\u3060\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

0\u306e\u968e\u4e57\u306f\uff1f<\/h2>\n\n\n
\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u3068\u3053\u308d\u3067\u30010\u306e\u968e\u4e57\u306f0\u306b\u306a\u308a\u307e\u3059\u304b\uff1f<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

0\u306e\u968e\u4e57\u306f1\u306b\u306a\u308a\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

0\u306e\u968e\u4e57<\/span><\/div>
\n

\\(0!=1\\)<\/p>\n<\/div><\/div>\n\n\n\n

\u610f\u5916\u306b\u611f\u3058\u305f\u65b9\u3082\u591a\u3044\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u304c\u3001\u968e\u4e57\u3092\u4e26\u3073\u66ff\u3048\u3068\u3057\u3066\u8003\u3048\u308b\u3068\u30010!\u306f0\u4eba\u306e\u4e26\u3073\u66ff\u3048\u306a\u306e\u3067\u300c\u4f55\u3082\u306a\u3044\u300d\u306e1\u901a\u308a<\/span>\u306b\u306a\u308b\u306e\u3067\u3001\\(0!\\)\u306f\\(1\\)\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u3059\u3053\u3057\u30e2\u30e4\u30e2\u30e4\u3057\u307e\u3059\u304c\u30010\u306e\u968e\u4e57\u306f1\u306b\u306a\u308b\u3053\u3068\u3092\u899a\u3048\u3066\u304a\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n

\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

\u968e\u4e57\u306f\u4e26\u3073\u66ff\u3048\u3092\u8868\u3057\u3066\u3044\u308b\u3093\u3060\u3063\u305f\u306d\uff01<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n

\"\"\u30b7\u30fc\u30bf<\/span><\/div>
\n

\u305d\u308c\u306a\u30890\u306e\u968e\u4e57\u304c1\u901a\u308a\u306a\u306e\u3082\u7d0d\u5f97\u3067\u304d\u308b\u304b\u306a<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u5206\u6570\u306e\u968e\u4e57\u306f\uff1f<\/h2>\n\n\n
\"\"\u9ad8\u6821\u751f<\/span><\/div>
\n

0\u306e\u968e\u4e57\u304c1\u306a\u306e\u306f\u5206\u304b\u3063\u305f\u3051\u3069\u3001\u5206\u6570\u306b\u968e\u4e57\u304c\u51fa\u3066\u304d\u305f\u3089\u3069\u3046\u3059\u308c\u3070\u3044\u3044\u3093\u3060\u308d\u3046\uff1f<\/p>\n<\/span><\/span><\/span><\/div><\/div><\/div><\/div>\n\n\n

\u5206\u6570\u306b\u968e\u4e57\u304c\u542b\u307e\u308c\u3066\u3044\u308b\u554f\u984c\u3082\u3042\u308a\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\u968e\u4e57\u3092\u542b\u3080\u5206\u6570<\/span><\/div>
\n

\\(\\displaystyle \\frac{5!}{6!} , \\frac{3!\\times{4!}}{12} , \\frac{n!}{(n+1)!}\\)<\/p>\n<\/div><\/div>\n\n\n\n

\u898b\u305f\u76ee\u306f\u96e3\u3057\u305d\u3046\u3067\u3059\u304c\u3001\u3053\u306e\u3088\u3046\u306a\u5206\u6570\u306f\u7d04\u5206\u3092\u3059\u308b\u3068\u304b\u3093\u305f\u3093\u306b\u306a\u308a\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\\(\\displaystyle \\frac{5!}{6!}=\\frac{5\\cdot4\\cdot3\\cdot2\\cdot1}{6\\cdot5\\cdot4\\cdot3\\cdot2\\cdot1}=\\frac{1}{6}\\)<\/p>\n\n\n\n

\\(\\displaystyle \\frac{3!\\times{4!}}{12}=\\frac{3\\cdot2\\cdot1 \\times 4\\cdot3\\cdot2\\cdot1}{12}=12\\)<\/p>\n\n\n\n

\\(\\displaystyle \\frac{n!}{(n+1)!}=\\frac{n\\cdot(n-1)\\cdot(n-2)\\cdots2\\cdot1}{(n+1)\\cdot n \\cdot (n-1)\\cdots 2\\cdot1 }=\\frac{1}{n+1}\\)<\/p>\n\n\n\n

\u6c7a\u3057\u3066\u3080\u305a\u304b\u3057\u3044\u8a08\u7b97\u3067\u306f\u306a\u3044\u306e\u3067\u3001\u843d\u3061\u7740\u3044\u3066\u53d6\u308a\u7d44\u3093\u3067\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u968e\u4e57\u306e\u548c\u3068\u5dee<\/h2>\n\n\n\n

\u968e\u4e57\u306e\u548c\u3068\u5dee\u3092\u6c42\u3081\u308b\u3068\u304d\u306f\u5de5\u592b\u304c\u5fc5\u8981\u3067\u3059\u3002<\/p>\n\n\n\n

\n

\\begin{eqnarray}
5!+4! &=& 5 \\cdot 4! +4! \\\\
&=& (5+1)\\times 4! \\\\
&=& 6 \\times 24 \\\\
&=& 144
\\end{eqnarray}\\begin{eqnarray}
6!-5! &=& 6 \\cdot 5! -5! \\\\
&=& (6-1) \\times 5!\\\\
&=& 5\\times 5! \\\\
&=&600
\\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

\u968e\u4e57\u306e\u5171\u901a\u90e8\u5206\u3092\u304f\u304f\u308a\u51fa\u3057\u3066\u304b\u3089\u8a08\u7b97\u3059\u308b<\/span>\u3068\u3001\u305d\u308c\u305e\u308c\u306e\u968e\u4e57\u3092\u8a08\u7b97\u3057\u3066\u304b\u3089\u548c\u3084\u5dee\u3092\u6c42\u3081\u308b\u3088\u308a\u305a\u3063\u3068\u697d\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u968e\u4e57\u306e\u8a08\u7b97\uff1c\u7df4\u7fd2\u554f\u984c\uff1e<\/h2>\n\n\n
\n
\"\u968e\u4e57\u306e\u8a08\u7b97\uff1c\u7df4\u7fd2\u554f\u984c\uff1e\"<\/figure>\n<\/div>\n\n\n

\u968e\u4e57\u306e\u8a08\u7b97\u3092\u3064\u304b\u3063\u305f\u7df4\u7fd2\u554f\u984c\u3092\u89e3\u304d\u307e\u3059\u3002<\/span><\/p>\n\n\n\n

\u7df4\u7fd2\u554f\u984c\uff11<\/span>4\u4eba\u306e\u751f\u5f92\u5168\u54e1\u30921\u5217\u306b\u4e26\u3079\u308b\u3068\u304d\u3001\u4e26\u3079\u65b9\u306e\u7dcf\u6570\u3092\u6c42\u3081\u3088\u3002<\/div>\n\n\n\n

\u89e3\u8aac<\/span><\/p>\n\n\n\n

4\u4eba\u5168\u54e1\u306e\u4e26\u3079\u65b9\u306e\u7dcf\u6570\u306a\u306e\u3067\u3001\u968e\u4e57\u3092\u3064\u304b\u3063\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

4\u4eba\u306e\u5217\u306a\u306e\u30674!\u3068\u306a\u308a\u3001<\/p>\n\n\n\n

\\(4!=4\\times{3}\\times{2}\\times{1}=24\\)<\/p>\n\n\n\n

\u3057\u305f\u304c\u3063\u3066\u300124\u901a\u308a\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u7df4\u7fd2\u554f\u984c\uff12<\/span>\u7537\u5b504\u4eba\u3068\u5973\u5b503\u4eba\u304c1\u5217\u306b\u4e26\u3076\u3068\u304d\u3001\u5973\u5b503\u4eba\u304c\u7d9a\u3044\u3066\u4e26\u3076\u4e26\u3073\u65b9\u306f\u4f55\u901a\u308a\u3042\u308b\u304b\u3002<\/div>\n\n\n\n

\u89e3\u8aac<\/span>
\u5973\u5b503\u4eba\u3092\u3072\u3068\u307e\u3068\u3081\u306b\u3057\u3066\u8003\u3048\u308b\u3002
\u7537\u5b504\u4eba\u3068\u5973\u5b50\u306e\u307e\u3068\u307e\u308a\u306e\u4e26\u3073\u65b9\u306f\u30015!\u901a\u308a\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u307e\u305f\u3001\u3072\u3068\u307e\u3068\u3081\u306b\u3057\u305f\u5973\u5b503\u4eba\u306e\u4e26\u3073\u65b9\u3082\u8003\u3048\u306a\u3051\u308c\u3070\u3044\u3051\u306a\u3044\u306e\u3067\u3001
\u5973\u5b503\u4eba\u306e\u4e26\u3073\u65b9\u306f3!\u901a\u308a\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u3088\u3063\u3066\u3001\u4e26\u3073\u65b9\u306e\u7dcf\u6570\u306f\u3001\u7a4d\u306e\u6cd5\u5247\u306b\u3088\u308a
\\(5!\\times{3!}=5\\times{4}\\times{3}\\times{2}\\times{1}\\times{3}\\times{2}\\times{1}=720\\)\u3057\u305f\u304c\u3063\u3066\u3001720\u901a\u308a\u3067\u3042\u308b\u3002<\/p>\n\n\n\n

\u968e\u4e57\u306e\u8a08\u7b97\u3000\u307e\u3068\u3081<\/h2>\n\n\n\n

\u4eca\u56de\u306f\u6570\u5b66A\u306e\u968e\u4e57\u306e\u8a08\u7b97\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n

\n
\n
\u968e\u4e57<\/dt>\n
1\u304b\u3089\\(n\\)\u307e\u3067\u306e\u3059\u3079\u3066\u306e\u81ea\u7136\u6570\u306e\u7a4d<\/dd>\n<\/dl>\n<\/div>\n\n\n\n
\u968e\u4e57\u306e\u8a08\u7b97
\\[n!=n \\cdot (n-1) \\cdot (n-2) \\cdots 2 \\cdot 1\\]<\/div>\n\n\n\n

\u968e\u4e57\u3092\u542b\u3080\u5206\u6570\u3084\u3001\u968e\u4e57\u306e\u548c\u3068\u5dee\u3082\u3057\u3063\u304b\u308a\u3068\u89e3\u3051\u308b\u3088\u3046\u306b\u3057\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\n

\\begin{eqnarray}
5!+4! &=& 5 \\cdot 4! +4! \\\\
&=& (5+1)\\times 4! \\\\
&=& 6 \\times 24 \\\\
&=& 144
\\end{eqnarray}<\/p>\n\n\n\n

\\begin{eqnarray}
6!-5! &=& 6 \\cdot 5! -5! \\\\
&=& (6-1) \\times 5!\\\\
&=&5\\times 5! \\\\
&=& 600
\\end{eqnarray}<\/p>\n<\/div><\/div>\n\n\n\n

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