(i)\u3000\\(1\\) \u307e\u305f\u306f \\(2\\) \u306e\u76ee\u304c\u51fa\u305f\u3068\u304d\u306b\u306f, Q \u3092 \\(x\\) \u8ef8\u306e\u6b63\u306e\u65b9\u5411\u306b \\(1\\) \u3060\u3051\u52d5\u304b\u3059.<\/p><\/li>\r\n
(ii)\u3000\\(3\\) \u307e\u305f\u306f \\(4\\) \u306e\u76ee\u304c\u51fa\u305f\u3068\u304d\u306b\u306f, Q \u3092 \\(y\\) \u8ef8\u306e\u6b63\u306e\u65b9\u5411\u306b \\(1\\) \u3060\u3051\u52d5\u304b\u3059.<\/p><\/li>\r\n
(iii)\u3000\\(5\\) \u307e\u305f\u306f \\(6\\) \u306e\u76ee\u304c\u51fa\u305f\u3068\u304d\u306b\u306f, Q \u3092\u52d5\u304b\u3055\u306a\u3044.<\/p><\/li>\r\n<\/ol>\r\n
Q \u306f\u6700\u521d\u539f\u70b9 \\(( 0 , 0 )\\) \u306b\u3042\u308b. \u3053\u306e\u3055\u3044\u3053\u308d\u3092 \\(( n+1 )\\) \u56de\u6295\u3052, Q \u304c\u901a\u3063\u305f\u70b9\uff08\u539f\u70b9\u304a\u3088\u3073 Q \u306e\u6700\u7d42\u4f4d\u7f6e\u306e\u70b9\u3092\u542b\u3080\uff09\u306e\u96c6\u5408\u3092 \\(S\\) \u3068\u3059\u308b. \u305f\u3060\u3057, \\(n\\) \u306f\u81ea\u7136\u6570\u3068\u3059\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
- \r\n
(1)<\/strong>\u3000\\(S\\) \u304c\u70b9 \\(( 1 , n-1 )\\) \u3092\u542b\u3080\u78ba\u7387\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n
(2)<\/strong>\u3000\\(S\\) \u304c\u9818\u57df \\(x+y \\lt n\\) \u306b\u542b\u307e\u308c\u308b\u78ba\u7387\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n
(3)<\/strong>\u3000\\(S\\) \u304c\u70b9 \\(( k , n-k )\\) \u3092\u542b\u3080\u306a\u3089\u3070\u5f97\u70b9 \\(2^k\\) \u70b9\uff08 \\(k = 0 , 1 , \\cdots , n\\) \uff09\u304c\u4e0e\u3048\u3089\u308c, \\(S\\) \u304c\u9818\u57df \\(x+y \\lt n\\) \u306b\u542b\u307e\u308c\u308b\u306a\u3089\u3070\u5f97\u70b9 \\(0\\) \u70b9\u304c\u4e0e\u3048\u3089\u308c\u308b\u3068\u3059\u308b. \u5f97\u70b9\u306e\u671f\u5f85\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
(1)<\/strong><\/p>\r\n
\\(1\\) \u56de\u3055\u3044\u3053\u308d\u3092\u6295\u3052\u305f\u3068\u304d\u306e Q \u306e\u52d5\u304d\u3068\u78ba\u7387\u306f\u4ee5\u4e0b\u306e\u3068\u304a\u308a.<\/p>\r\n
![\"yokokoku2010_02_01\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/yokokoku2010_02_01.png\")
\r\n\u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u306f\u4ee5\u4e0b\u306e \\(2\\) \u30d1\u30bf\u30fc\u30f3\u304c\u3042\u308b.<\/p>\r\n
- \r\n
1*<\/strong>\u3000Q \u304c\u70b9 \\(( 1 , n )\\) \u304b\u70b9 \\(( 2 , n-1 )\\) \u3067\u7d42\u308f\u308b.<\/p><\/li>\r\n
2*<\/strong>\u3000Q \u304c\u70b9 \\(( 1 , n-1 )\\) \u3067\u7d42\u308f\u308b.<\/p><\/li>\r\n<\/ol>\r\n
- \r\n
1*<\/strong> \u3068\u306a\u308b\u78ba\u7387
\r\n\\(n\\) \u56de\u76ee\u307e\u3067\u306b, \\(1\\) \u304b \\(2\\) \u304c \\(1\\) \u56de, \\(3\\) \u304b \\(4\\) \u304c \\(n-1\\) \u56de\u51fa\u3066, \\(n+1\\) \u56de\u76ee\u3067, \\(1\\) \uff5e \\(4\\) \u304c\u51fa\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\r\n{} _ n \\text{C} {} _ 1 p \\cdot p^{n-1} \\cdot 2p = 2np^{n+1} \\quad ... [1]\r\n\\]<\/li>\r\n2*<\/strong> \u3068\u306a\u308b\u78ba\u7387
\r\n\\(n+1\\) \u56de\u76ee\u307e\u3067\u306b, \\(1\\) \u304b \\(2\\) \u304c \\(1\\) \u56de, \\(3\\) \u304b \\(4\\) \u304c \\(n-1\\) \u56de, \\(5\\) \u304b \\(6\\) \u304c \\(1\\) \u56de\u51fa\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\r\n\\dfrac{(n+1)!}{1! 1! (n-1)!} p \\cdot p^{n-1} \\cdot (1-2p) = n(n+1) (1-2p) p^n \\quad ... [2]\r\n\\]<\/li>\r\n<\/ol>\r\n[1] [2] \u3088\u308a, \u6c42\u3081\u308b\u78ba\u7387\u306f\r\n\\[\r\n2np^{n+1} + n(n+1) (1-2p) p^n = \\underline{n \\{1+n(1-2p)\\} p^n}\r\n\\]\r\n(2)<\/strong><\/p>\r\n
\\(5\\) \u304b \\(6\\) \u304c\u5c11\u306a\u304f\u3068\u3082 \\(2\\) \u56de\u51fa\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=2}^{n+1} & {} _ {n+1} \\text{C} {} _ k (1-2p)^k (2p)^{n+1-k} \\\\\r\n& = \\left\\{ (1-2p) +2p \\right\\}^{n+1} -(2p)^{n+1} -{} _ {n+1} \\text{C} {} _ 1 (1-2p) (2p)^n \\\\\r\n& = 1 -(2p)^{n+1} -(n+1)(1-2p) (2p)^n \\\\\r\n& = 1 -\\{ 2p +(n+1)(1-2p) \\}(2p)^n \\\\\r\n& = \\underline{1 -\\{1+n(1-2p)\\}(2p)^n}\r\n\\end{align}\\]\r\n
(3)<\/strong><\/p>\r\n
\\(S\\) \u304c\u70b9 \\(( k , n-k )\\) \u3092\u542b\u3080\u78ba\u7387\u3092, (1)<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308b.
\r\n\u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u306f\u4ee5\u4e0b\u306e \\(2\\) \u30d1\u30bf\u30fc\u30f3\u304c\u3042\u308b.<\/p>\r\n- \r\n
1*<\/strong>\u3000Q \u304c\u70b9 \\(( k+1 , n-k )\\) \u304b\u70b9 \\(( k , n+1-k )\\) \u3067\u7d42\u308f\u308b.<\/p><\/li>\r\n
2*<\/strong>\u3000Q \u304c\u70b9 \\(( k , n-k )\\) \u3067\u7d42\u308f\u308b.<\/p><\/li>\r\n<\/ol>\r\n
- \r\n
1*<\/strong> \u3068\u306a\u308b\u78ba\u7387\u306f\r\n\\[\r\n{} _ n \\text{C} {} _ k p^k \\cdot p^{n-k} \\cdot 2p = 2 {} _ n \\text{C} {} _ k p^{n+1} \\quad ... [3]\r\n\\]<\/li>\r\n
2*<\/strong> \u3068\u306a\u308b\u78ba\u7387\u306f\r\n\\[\r\n\\dfrac{(n+1)!}{1! k! (n-k)!} p^k \\cdot p^{n-k} \\cdot (1-2p) = (n+1) {} _ n \\text{C} {} _ k (1-2p) p^n \\quad ... [4]\r\n\\]<\/li>\r\n<\/ol>\r\n[3] [4] \u3088\u308a\r\n\\[\r\n2 {} _ n \\text{C} {} _ k p^{n+1} +(n+1) {} _ n \\text{C} {} _ k (1-2p) p^n = {} _ n \\text{C} {} _ k \\{1+n(1-2p)\\} p^n\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070, \u6c42\u3081\u308b\u671f\u5f85\u5024\u306f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=0}^{n} & 2^k \\cdot {} _ n \\text{C} {} _ k \\{1+n(1-2p)\\} p^n \\\\\r\n& = \\{1+n(1-2p)\\} p^n \\cdot \\textstyle\\sum\\limits _ {k=0}^{n} {} _ n \\text{C} {} _ k 2^k \\cdot 1^{n-k} \\\\\r\n& = \\{1+n(1-2p)\\} p^n \\cdot (2+1)^n \\\\\r\n& = \\underline{\\{1+n(1-2p)\\} (3p)^n}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(1\\) \u500b\u306e\u3044\u3073\u3064\u306a\u3055\u3044\u3053\u308d\u304c\u3042\u308b. \\(1, 2, 3, 4\\) \u306e\u76ee\u304c\u51fa\u308b\u78ba\u7387\u306f\u305d\u308c\u305e\u308c \\(\\dfrac{p}{2}\\) \u3067\u3042\u308a, \\(5, 6\\) \u306e\u76ee\u304c\u51fa\u308b\u78ba\u7387\u306f\u305d\u308c\u305e\u308c \\(\\dfrac{1-2p}{2} … \u7d9a\u304d\u3092\u8aad\u3080