从 1～n 中取 k 个不同的数相加求和，求这个和的末位数等于 0,1,2,…,9 的频数

lihp2020

Num8偶=1*215553195+990*8145060+148995*148995+8145060*990+215553195*1 =38757835215
Num8奇=45*45379620+14190*1221759+1221759*14190+45379620*45= 38757686220
验证 C(90,8)=77515521435=38757835215+38757686220

Num6偶=1*8145060+990*148995+148995*990+8145060*1 =311300220
Num6奇=45*1221759+14190*14190+1221759*45= 311314410
验证 C(90,6)=622614630=311300220+311314410

结果余数是奇数 12579
C（90,9）/10 +Num8奇/5 + Num8偶/5*2 + C（90,7）/10*3 +Num6奇/5
=706252528630/10+38757686220/5+38757835215/5*2+7471375560/10*3+311314410/5
=96183599743
余数是偶数 02468 一共有
C（90,9）/10 +Num8偶/5 + Num8奇/5*2 + C（90,7）/10*3 +Num6偶/5
=706252528630/10+38757835215/5+38757686220/5*2+7471375560/10*3+311300220/5
=96183567106

验证  96183599743*5+96183567106*5=961835834245 =C(93,9)

王守恩

谢谢 cooooldog, 谢谢 luyuanhong, 谢谢 天山草。

谢谢 northwolves 给出解题模块(93,9,Mod10 都可以改)

1. n = 93; m = 9;
   
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]},
   
5.       {x, Permutations@b[[k]]}], First -> Last, Total], {k, 1, Length[b]}];
   
6. d = Normal@Merge[c, Total] // Sort;
   
7. e = {Total@Values[d], d}

复制代码

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

谢谢楼主！给我们出了道好题。

王守恩

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(公式在22楼)
   {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(主帖)
   {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,......