通项公式

northwolves

1. n = 365; m = 12;
   
2. a = Table[Count[Mod[Range@n, 10], m], {m, 0, 9}];
   
3. b = PowersRepresentations[m, 10, 1];
   
4. c = Table[GroupBy[Table[{Mod[Total[x*Range[0, 9]], 10], Times @@ Binomial[a, x]}, {x, Permutations@b[[k]]}], First -> Last, Total], {k, 1, Length[b]}];
   
5. d = Normal@Merge[c, Total] // Sort;
   
6. e = {Total@Values[d], d}

复制代码

{9723205992282927449305, {0 -> 972320599232937216127,
  1 -> 972320599223648273806, 2 -> 972320599232937216127,
  3 -> 972320599223648273770, 4 -> 972320599232937216091,
  5 -> 972320599223648273734, 6 -> 972320599232937216055,
  7 -> 972320599223648273734, 8 -> 972320599232937216091,
  9 -> 972320599223648273770}}

王守恩

从1到365这365个自然数中任取两两不等的12个数, 求取到 12 个数之和恰好等于 2023 有多少组？

mathe

1~n中选择m个不同的数，求和为S的不同方案，可以采用动态规划，设
A[k,h,s]表示从1～k中选择h个数，和为s的方案数目
于是
A[k+1,h,s]=A[k,h,s]+A[k,h-1,s-(k+1)]。
利用这个公式就可以递推计算了。

另外一方面我们也可以进行估算。
1～n中选择一个数，平均值为$e=(1+n)/2$.
而方差为$V=1/n((1-e)^2+...+(n-e)^2)=(n+1)(2n+1)/6-(n+1)^2/4=(n^2-1)/12$
而两次不同选择不能相等，所以它们的协方差为$E(xi-e)(xj-e)=-V/{n-1}$
所以$E(x_1+x_2+...+x_m)=m e, Var(x_1+x_2+...+x_m)=E((\sum(x_i-e))^2)=mV+m(m-1)*(-V)/{n-1}=\frac{m(n-m)V}{n-1}$
也就是我们可以估计均值为$m e$, 方差为$\frac{m(n-m)V}{n-1}$的正态分母，和在$(S-1/2,S+1/2]$中的概率作为估计值。
比如n=365,m=12, e=183,V=11102,对应最后和值的均值为2196,方差129198.S=2023
也就是要求求标准正态分布在区间(-0.48269364796817,-0.47991155201446]上的概率。然后将概率乘上数目n(n-1)(n-2)...(n-m+1)即可

王守恩

从 1~365 中取 12 个不同数, 求取到 12 个数之和恰好等于 2023 有 9,542,858,902,052,954,887 组。

CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{353}\frac{x^{366}-x^i}{x^{i+177}-x^{177}},(x,0,4236)\bigg],x\bigg]\)

王守恩

从1~365中取2个不同数, 2个数之和恰好等于366有182组。(366是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{363}\frac{x^{366}-x^i}{x^{i+182}-x^{182}},(x,0,364)\bigg],x\bigg]\)[[364]]

从1~365中取12个不同数, 12个数之和恰好等于2196有 10,655,873,125,881,071,133组。(2196是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{353}\frac{x^{366}-x^i}{x^{i+177}-x^{177}},(x,0,2119)\bigg],x\bigg]\)[[2119]]

从1~365中取22个不同数, 22个数之和恰好等于4026有 90,400,006,783,955,844,446,363,021,686,652组。(4026是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{343}\frac{x^{366}-x^i}{x^{i+172}-x^{172}},(x,0,3774)\bigg],x\bigg]\)[[3774]]

从1~365中取32个不同数, 32个数之和恰好等于5856有 6,429,619,212,767,264,081,938,214,593,840,653,588,467,393组。(5856是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{333}\frac{x^{366}-x^i}{x^{i+167}-x^{167}},(x,0,5329)\bigg],x\bigg]\)[[5329]]

从1~365中取42个不同数, 42个数之和恰好等于7686有 15,628,610,385,975,443,405,198,520,232,956,659,310,953,492,784,582,040组。(7686是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{323}\frac{x^{366}-x^i}{x^{i+162}-x^{162}},(x,0,6784)\bigg],x\bigg]\)[[6784]]

从1~365中取52个不同数, 52个数之和恰好等于9516有 2,670,630,333,002,279,960,845,281,844,182,378,854,847,648,865,674,133,712,853,542组。(9516是这串数里最大的1个)
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{313}\frac{x^{366}-x^i}{x^{i+157}-x^{157}},(x,0,8139)\bigg],x\bigg]\)[[8139]]

从1~365中取62个不同数, 62个数之和恰好等于11346有 49,748,750,626,982,865,763,986,416,940,067,779,162,302,652,366,469,638,594,637,743,082,250组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{303}\frac{x^{366}-x^i}{x^{i+152}-x^{152}},(x,0,9394)\bigg],x\bigg]\)[[9394]]

从1~365中取72个不同数, 72个数之和恰好等于13176有 135,481,907,930,175,289,355,266,190,568,641,679,304,610,330,180,572,889,020,549,945,144,295,033,233组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{293}\frac{x^{366}-x^i}{x^{i+147}-x^{147}},(x,0,10549)\bigg],x\bigg]\)[[10549]]

从1~365中取82个不同数, 82个数之和恰好等于15006有 66,450,206,598,429,682,628,568,057,932,527,035,042,767,873,072,888,187,003,576,384,540,964,503,149,956,526组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{283}\frac{x^{366}-x^i}{x^{i+142}-x^{142}},(x,0,11604)\bigg],x\bigg]\)[[11604]]

从1~365中取92个不同数, 92个数之和恰好等于16836有 6,849,865,466,092,346,029,128,136,216,200,617,602,021,866,965,346,024,560,761,533,508,061,452,355,694,620,310,906组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{273}\frac{x^{366}-x^i}{x^{i+137}-x^{137}},(x,0,12559)\bigg],x\bigg]\)[[12559]]

从1~365中取102个不同数,10 2个数之和恰好等于18666有 166,906,812,106,900,780,203,168,662,504,043,192,206,214,971,270,047,654,796,119,609,935,328,361,382,023,100,614,735,540组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{263}\frac{x^{366}-x^i}{x^{i+132}-x^{132}},(x,0,13414)\bigg],x\bigg]\)[[13414]]

从1~365中取112个不同数, 112个数之和恰好等于20496有 1,053,013,743,772,085,687,318,774,904,578,338,951,685,097,138,572,258,364,487,025,174,930,071,037,936,299,865,479,215,841,832组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{253}\frac{x^{366}-x^i}{x^{i+127}-x^{127}},(x,0,14169)\bigg],x\bigg]\)[[14169]]

从1~365中取122个不同数, 122个数之和恰好等于22326有 1,847,357,036,944,212,468,060,799,371,115,931,445,420,684,557,057,887,411,515,479,980,317,488,280,984,252,706,743,454,967,247,070组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{243}\frac{x^{366}-x^i}{x^{i+122}-x^{122}},(x,0,14824)\bigg],x\bigg]\)[[14824]]

从1~365中取132个不同数, 132个数之和恰好等于24156有 953,163,733,503,859,111,013,051,616,701,688,309,768,626,876,182,270,382,208,477,563,894,791,704,733,584,473,982,505,746,826,001,976组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{233}\frac{x^{366}-x^i}{x^{i+117}-x^{117}},(x,0,15379)\bigg],x\bigg]\)[[15379]]

从1~365中取142个不同数, 142个数之和恰好等于25986有 151,111,644,158,877,461,615,176,016,165,544,111,064,863,470,331,198,744,416,850,124,028,633,790,549,084,960,148,265,478,359,884,017,320组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{223}\frac{x^{366}-x^i}{x^{i+112}-x^{112}},(x,0,15834)\bigg],x\bigg]\)[[15834]]

从1~365中取152个不同数, 152个数之和恰好等于27816有 7,612,758,755,330,028,426,971,377,342,459,599,737,864,438,968,185,822,628,924,554,168,720,288,895,463,943,491,508,406,071,720,893,135,890组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{213}\frac{x^{366}-x^i}{x^{i+107}-x^{107}},(x,0,16189)\bigg],x\bigg]\)[[16189]]

从1~365中取162个不同数, 162个数之和恰好等于29646有 124,946,702,870,321,967,235,134,974,592,997,204,874,275,672,831,335,078,368,012,511,567,260,443,497,143,702,125,544,961,809,200,467,128,672组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{203}\frac{x^{366}-x^i}{x^{i+102}-x^{102}},(x,0,16444)\bigg],x\bigg]\)[[16444]]

从1~365中取172个不同数, 172个数之和恰好等于31476有 679,728,004,010,725,708,247,275,762,250,148,714,810,033,837,556,572,821,688,774,046,128,985,369,426,971,236,533,144,606,302,636,446,352,536组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{193}\frac{x^{366}-x^i}{x^{i+97}-x^{97}},(x,0,16599)\bigg],x\bigg]\)[[16599]]

从1~365中取182个不同数, 182个数之和恰好等于33306有 1,238,265,005,128,257,960,854,030,011,771,898,020,954,511,596,775,351,614,116,958,454,583,975,724,255,436,480,191,559,141,334,272,573,713,844组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{183}\frac{x^{366}-x^i}{x^{i+92}-x^{92}},(x,0,16654)\bigg],x\bigg]\)[[16654]]

从1~365中取192个不同数, 192个数之和恰好等于35136有 758,081,700,107,993,035,974,870,570,958,151,684,770,830,178,845,299,115,482,249,293,474,685,990,743,934,031,053,607,491,130,986,299,383,766组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{173}\frac{x^{366}-x^i}{x^{i+87}-x^{87}},(x,0,16609)\bigg],x\bigg]\)[[16609]]

从1~365中取202个不同数, 202个数之和恰好等于36966有 155,514,236,933,631,890,218,139,215,530,080,854,070,079,724,023,883,542,506,428,889,829,152,951,912,499,151,127,994,560,155,297,814,827,366组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{163}\frac{x^{366}-x^i}{x^{i+82}-x^{82}},(x,0,16464)\bigg],x\bigg]\)[[16464]]

从1~365中取212个不同数, 212个数之和恰好等于38796有 10,588,380,801,539,873,124,471,285,828,821,229,065,127,060,067,905,073,897,150,999,733,500,302,048,969,024,466,486,639,266,713,969,943,140组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{153}\frac{x^{366}-x^i}{x^{i+77}-x^{77}},(x,0,16219)\bigg],x\bigg]\)[[16219]]

从1~365中取222个不同数, 222个数之和恰好等于40626有 235,353,548,961,557,518,090,265,855,544,362,566,381,076,303,491,753,610,150,198,803,572,512,308,695,326,680,782,939,221,354,363,049,912组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{143}\frac{x^{366}-x^i}{x^{i+72}-x^{72}},(x,0,15874)\bigg],x\bigg]\)[[15874]]

从1~365中取232个不同数, 232个数之和恰好等于42456有 1,667,129,803,153,419,350,812,012,857,889,233,121,847,924,396,907,857,590,349,580,906,332,765,313,443,023,086,451,903,964,492,837,064组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{133}\frac{x^{366}-x^i}{x^{i+67}-x^{67}},(x,0,15429)\bigg],x\bigg]\)[[15429]]

从1~365中取242个不同数, 242个数之和恰好等于44286有 3,642,319,656,811,713,231,479,712,181,723,377,007,094,138,968,197,459,573,160,046,031,199,900,221,975,544,688,935,092,012,191,512组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{123}\frac{x^{366}-x^i}{x^{i+62}-x^{62}},(x,0,14884)\bigg],x\bigg]\)[[14884]]

从1~365中取252个不同数, 252个数之和恰好等于46116有 2,351,852,253,191,729,120,650,288,329,465,613,792,778,715,770,215,355,092,264,103,255,736,054,761,882,201,199,657,550,173,512组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{113}\frac{x^{366}-x^i}{x^{i+57}-x^{57}},(x,0,14239)\bigg],x\bigg]\)[[14239]]

从1~365中取262个不同数, 262个数之和恰好等于47946有 424,919,401,107,490,493,518,326,367,912,045,983,124,343,990,522,312,515,326,340,333,333,114,957,222,134,687,397,253,984组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{103}\frac{x^{366}-x^i}{x^{i+52}-x^{52}},(x,0,13494)\bigg],x\bigg]\)[[13494]]

从1~365中取272个不同数, 272个数之和恰好等于49776有 20,036,313,201,925,393,250,559,788,653,853,214,948,002,538,959,790,992,550,434,572,106,534,584,361,520,820,599,658组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{93}\frac{x^{366}-x^i}{x^{i+47}-x^{47}},(x,0,12649)\bigg],x\bigg]\)[[12649]]

从1~365中取282个不同数, 282个数之和恰好等于51606有 225,605,670,846,057,198,414,201,222,180,316,675,053,629,232,373,832,083,827,237,812,946,676,964,835,637,830组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{83}\frac{x^{366}-x^i}{x^{i+42}-x^{42}},(x,0,11704)\bigg],x\bigg]\)[[11704]]

从1~365中取292个不同数, 292个数之和恰好等于53436有 540,984,660,192,891,612,140,640,600,117,334,181,790,325,017,800,698,031,840,796,058,589,909,220,173组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{73}\frac{x^{366}-x^i}{x^{i+37}-x^{37}},(x,0,10659)\bigg],x\bigg]\)[[10659]]

从1~365中取302个不同数, 302个数之和恰好等于55266有 237,762,695,973,447,690,351,046,491,893,641,153,195,144,866,856,706,230,388,026,059,555,530组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{63}\frac{x^{366}-x^i}{x^{i+32}-x^{32}},(x,0,9514)\bigg],x\bigg]\)[[9514]]

从1~365中取312个不同数, 312个数之和恰好等于57096有 15,648,184,566,422,670,867,196,470,627,366,929,481,481,374,672,851,806,953,546,944组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{53}\frac{x^{366}-x^i}{x^{i+27}-x^{27}},(x,0,8269)\bigg],x\bigg]\)[[8269]]

从1~365中取322个不同数, 322个数之和恰好等于58926有 116,212,775,733,205,156,816,065,702,872,450,113,210,545,218,239,534,542组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{43}\frac{x^{366}-x^i}{x^{i+22}-x^{22}},(x,0,6924)\bigg],x\bigg]\)[[6924]]

从1~365中取332个不同数, 332个数之和恰好等于60756有 63,995,334,301,723,685,441,535,021,060,219,672,927,820,493组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{33}\frac{x^{366}-x^i}{x^{i+17}-x^{17}},(x,0,5479)\bigg],x\bigg]\)[[5479]]

从1~365中取342个不同数, 342个数之和恰好等于62886有 1,320,827,419,936,988,491,388,265,553,443,322组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{23}\frac{x^{366}-x^i}{x^{i+12}-x^{12}},(x,0,3934)\bigg],x\bigg]\)[[3934]]

从1~365中取352个不同数,352个数之和恰好等于64416有 278,664,877,516,110,742,675组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{13}\frac{x^{366}-x^i}{x^{i+7}-x^{7}},(x,0,2289)\bigg],x\bigg]\)[[2289]]

从1~365中取362个不同数,362个数之和恰好等于66246有 16,562组。
CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\prod_{i=1}^{3}\frac{x^{366}-x^i}{x^{i+2}-x^{2}},(x,0,544)\bigg],x\bigg]\)[[544]]

王守恩

1,0,5,0,9,0,13,0,17,0,21,0,25,0,29,......

northwolves

1. Riffle[Range@10*4 - 3, 0]

复制代码

王守恩

谢谢northwolves！分享这么个好东西！ 谢谢Treenewbee！

Table\(\bigg[\)CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\frac{1 + 3 x^n}{(1 - x^n)^2}, {x, 0, 40 n}\bigg], x\bigg], {n, 1, 5}\bigg]\)

{1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69,73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129,133, 137, 141, 145, 149, 153, 157, 161},
{1, 0, 5, 0, 9, 0, 13, 0, 17, 0, 21, 0, 25, 0, 29, 0, 33, 0, 37, 0, 41, 0, 45, 0, 49, 0, 53, 0, 57, 0, 61, 0, 65, 0, 69, 0, 73, 0, 77, 0, 81, 0, 85, 0, 89, 0, 93, 0, 97, 0, 101, 0, ......
{1, 0, 0, 5, 0, 0, 9, 0, 0, 13, 0, 0, 17, 0, 0, 21, 0, 0, 25,  0, 0, 29, 0, 0, 33, 0, 0, 37, 0, 0, 41, 0, 0, 45, 0, 0, 49, 0, 0, 53, 0, 0, 57, 0, 0, 61, 0, 0, 65, 0, 0, 69, 0, 0, 73, ......
{1, 0, 0, 0, 5, 0, 0, 0, 9, 0, 0, 0, 13, 0, 0, 0, 17, 0, 0, 0, 21, 0, 0, 0, 25, 0, 0, 0, 29, 0, 0, 0, 33, 0, 0, 0, 37, 0, 0, 0, 41, 0, 0, 0, 45, 0, 0, 0, 49, 0, 0, 0, 53, 0, 0, 0, ......
{1, 0, 0, 0, 0, 5, 0, 0, 0, 0, 9, 0, 0, 0, 0, 13, 0, 0, 0, 0, 17, 0, 0, 0, 0, 21, 0, 0, 0, 0, 25, 0, 0, 0, 0, 29, 0, 0, 0, 0, 33, 0, 0, 0, 0, 37, 0, 0, 0, 0, 41, 0, 0, 0, 0, 45, ......

Table\(\bigg[\)CoefficientList\(\bigg[\)Series\(\bigg[\)\(\D\frac{1}{1 - x^n - x^{2n}}, {x, 0, 24 n}\bigg], x\bigg], {n, 1, 4}\bigg]\)

{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025},
{1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 8, 0, 13, 0, 21, 0, 34, 0, 55, 0, 89, 0, 144, 0, 233, 0, 377, 0, 610, 0, 987, 0, 1597, 0, 2584, 0, 4181, 0, 6765, 0, 10946, 0, 17711, 0, 28657, 0, 46368, 0, 75025},
{1, 0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 5, 0, 0, 8, 0, 0, 13, 0, 0, 21, 0, 0, 34, 0, 0, 55, 0, 0, 89, 0, 0, 144, 0, 0, 233, 0, 0, 377, 0, 0, 610, 0, 0, 987, 0, 0, 1597, 0, 0, 2584, 0, 0, 4181, 0, 0, ......
{1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 8, 0, 0, 0, 13, 0, 0, 0, 21, 0, 0, 0, 34, 0, 0, 0, 55, 0, 0, 0, 89, 0, 0, 0, 144, 0, 0, 0, 233, 0, 0, 0, 377, 0, 0, 0, 610, 0, 0, 0, 987, ........
  

王守恩

谢谢 northwolves！利用21楼的公式。

从1到93这93个自然数中任取两两不等的9个数, 把它们相加得到一个和;

根据模2
   {961835834245, {0 -> 480917835530, 1 -> 480917998715}}

根据模3
   {961835834245, {0 -> 320611969669, 1 -> 320611932288, 2 -> 320611932288}}

根据模4
   {961835834245, {0 -> 240458928561, 1 -> 240459018024, 2 -> 240458906969, 3 -> 240458980691}}

根据模5
   {961835834245, {0 -> 192367166849, 1 -> 192367166849, 2 -> 192367166849, 3 -> 192367166849, 4 -> 192367166849}}

根据模6
   {961835834245, {0 -> 160305833348, 1 -> 160306172766, 2 -> 160306242660, 3 -> 160306136321, 4 -> 160305759522, 5 -> 160305689628}}

根据模7
   {961835834245, {0 -> 137404699655, 1 -> 137405071093, 2 -> 137405480090, 3 -> 137405615242, 4 -> 137405378330,  5 -> 137404945807, 6 -> 137404644028}}

根据模8
   {961835834245, {0 -> 120229471007, 1 -> 120229496813, 2 -> 120229429363, 3 -> 120229467724, 4 -> 120229457554, 5 -> 120229521211, 6 -> 120229477606, 7 -> 120229512967}}

根据模9
   {961835834245, {0 -> 106870656520, 1 -> 106870644090, 2 -> 106870644090, 3 -> 106870656555, 4 -> 106870644117,
    5 -> 106870644117, 6 -> 106870656594, 7 -> 106870644081,  8 -> 106870644081}}

根据模10
   {961835834245, {0 -> 96183567106, 1 -> 96183599743, 2 -> 96183567106, 3 -> 96183599743, 4 -> 96183567106,
    5 -> 96183599743, 6 -> 96183567106,7 -> 96183599743, 8 -> 96183567106, 9 -> 96183599743}}

根据模11
   {961835834245, {0 -> 87439624480, 1 -> 87439621477, 2 -> 87439597167, 3 -> 87439621477, 4 -> 87439634347,
    5 -> 87439634347, 6 -> 87439634347, 7 -> 87439615042, 8 -> 87439615042, 9 -> 87439615042, 10 -> 87439621477}}

根据模12
   {961835834245, {0 -> 80152963488, 1 -> 80153371017, 2 -> 80153556426,  3 -> 80153543621, 4 -> 80153278839, 5 -> 80153054307,
    6 -> 80152869860, 7 -> 80152801749, 8 -> 80152686234, 9 -> 80152592700, 10 -> 80152480683, 11 -> 80152635321}}

根据模13
   {961835834245, {0 -> 73993597309, 1 -> 73992716506, 2 -> 73990737364, 3 -> 73988166730, 4 -> 73985592279, 5 -> 73983322066, 6 -> 73981828354,
    7 -> 73981380952, 8 -> 73982359830,  9 -> 73984624114,10 -> 73987750534, 11 -> 73990809535, 12 -> 73992948672}}

根据模14
   {961835834245, {0 -> 68713636855, 1 -> 68698889469, 2 -> 68684857015, 3 -> 68674272652, 4 -> 68669092888, 5 -> 68670535932, 6 -> 68678314683,
    7 -> 68691062800, 8 -> 68706181624,  9 -> 68720623075, 10 -> 68731342590,11 -> 68736285442, 12 -> 68734409875, 13 -> 68726329345}}

根据模15
   {961835834245, {0 -> 64039372505, 1 -> 64013017320, 2 -> 64005644223, 3 -> 64018367801, 4 -> 64049154318, 5 -> 64092579840, 6 -> 64141264565,
    7 -> 64186635357, 8 -> 64220888259, 9 -> 64238077529, 10 -> 64235214504, 11 -> 64212884964, 12 -> 64174887269, 13 -> 64127910789, 14 -> 64079935002}}
......

其中余数各不相同的是模2,4,5,6,7,8,12,13,14,15,16,17,18,19,20,21,22,23,......好像没有规律？

25楼：这串数已经有了，就不能按模分一下？

王守恩

从1到365这365个自然数中任取两两不等的2个数相加,  这个和的末位数可能是0,1,2,3,...,7,8,9,  求这十种情况出现的频数各是多少?
  {66430, {0 -> 6624, 1 -> 6660, 2 -> 6624, 3 -> 6661, 4 -> 6625, 5 -> 6662, 6 -> 6626, 7 -> 6662, 8 -> 6625, 9 -> 6661}}

从1到365这365个自然数中任取两两不等的12个数相加,  这个和的末位数可能是0,1,2,3,...,7,8,9,  求这十种情况出现的频数各是多少?
   {9723205992282927449305,  {0 -> 972320599232937216127, 1 -> 972320599223648273806, 2 -> 972320599232937216127,
  3 -> 972320599223648273770, 4 -> 972320599232937216091, 5 -> 972320599223648273734, 6 -> 972320599232937216055,
  7 -> 972320599223648273734, 8 -> 972320599232937216091, 9 -> 972320599223648273770}}

从1到365这365个自然数中任取两两不等的22个数相加,  这个和的末位数可能是0,1,2,3,...,7,8,9,  求这十种情况出现的频数各是多少?
   {109461191230675602553392425823040200,  {0 -> 10946119123067560241979823789249000, 1 -> 10946119123067560268698661375357780,
  2 -> 10946119123067560241979823789249000, 3 -> 10946119123067560268698661375358410, 4 -> 10946119123067560241979823789249630,
  5 -> 10946119123067560268698661375359040, 6 -> 10946119123067560241979823789250260, 7 -> 10946119123067560268698661375359040,
  8 -> 10946119123067560241979823789249630, 9 -> 10946119123067560268698661375358410}}

从1到365这365个自然数中任取两两不等的32个数相加,  这个和的末位数可能是0,1,2,3,...,7,8,9,  求这十种情况出现的频数各是多少?
   {9231778763124791646035107689625616155991385715,  {0 -> 923177876312479164603514281987056056848176119, 1 -> 923177876312479164603507255938067174350115304,
  2 -> 923177876312479164603514281987056056848176119, 3 -> 923177876312479164603507255938067174350108164, 4 -> 923177876312479164603514281987056056848168979,
  5 -> 923177876312479164603507255938067174350101024, 6 -> 923177876312479164603514281987056056848161839, 7 -> 923177876312479164603507255938067174350101024,
  8 -> 923177876312479164603514281987056056848168979, 9 -> 923177876312479164603507255938067174350108164}}

从1到365这365个自然数中任取两两不等的42个数相加,  这个和的末位数可能是0,1,2,3,...,7,8,9,  求这十种情况出现的频数各是多少?

从1到365这365个自然数中任取两两不等的52个数相加,  这个和的末位数可能是0,1,2,3,...,7,8,9,  求这十种情况出现的频数各是多少?