Hướng dẫn cho Hệ số nhị thức
Chỉ sử dụng khi thực sự cần thiết như một cách tôn trọng tác giả và người viết hướng dẫn này.
Chép code từ bài hướng dẫn để nộp bài là hành vi có thể dẫn đến khóa tài khoản.
Chép code từ bài hướng dẫn để nộp bài là hành vi có thể dẫn đến khóa tài khoản.
Authors:
Subtask \(1\):
Tutorial
Sử dụng công thức: \(\displaystyle \binom{n}{k} = \frac{n!}{(n - k)! \cdot k!}\).
Độ phức tạp: \(O(n)\).
Subtask \(2\):
Tutorial
Sử dụng tam giác Pascal: \(\displaystyle \binom{n}{k} = \binom{n - 1}{k} + \binom{n - 1}{k - 1}\).
Độ phức tạp: \(O(n \cdot k)\).
Subtask \(3\):
Solution
C++
C++
#include <bits/stdc++.h>
using namespace std;
const int mod = 1e9 + 7;
int factorial(int n) {
int res = 1;
for (int i = 1; i <= n; i++) {
res = res * (long long)i % mod;
}
return res;
}
int inverse(int x) {
if (x <= 1) {
return 1;
}
return (mod - mod / x) * (long long)inverse(mod % x) % mod;
}
int choose(int n, int k) {
if (n < 0 || k < 0 || n < k) {
return 0;
}
return factorial(n) * (long long)inverse(factorial(n - k) * (long long)factorial(k) % mod) % mod;
}
int main() {
int n, k;
cin >> n >> k;
cout << choose(n, k) << "\n";
return 0;
}
Pascal
Delphi
const
modulo = 1000000007;
function factorial(n: longint): int64;
var
i: longint;
res: int64;
begin
res := 1;
for i := 1 to n do
begin
res := res * i mod modulo;
end;
exit(res);
end;
function inverse(x: longint): int64;
begin
if x <= 1 then
begin
exit(1);
end;
exit((modulo - modulo div x) * inverse(modulo mod x) mod modulo);
end;
function choose(n, k: longint): int64;
begin
if (n < 0) or (k < 0) or (n < k) then
begin
exit(0);
end;
exit(factorial(n) * inverse(factorial(n - k) * factorial(k) mod modulo) mod modulo);
end;
var
n, k: longint;
begin
readln(n, k);
writeln(choose(n, k));
end.
Python
Python
mod = 10**9 + 7
def factorial(n):
res = 1
for i in range(1, n + 1):
res = res * i % mod
return res
def inverse(x):
if x <= 1:
return 1
return (mod - mod // x) * inverse(mod % x) % mod
def choose(n, k):
if n < 0 or k < 0 or n < k:
return 0
return factorial(n) * inverse(factorial(n - k) * factorial(k) % mod) % mod
n, k = map(int, input().split())
print(choose(n, k))
Subtask \(4\):
Tutorial
Ta sẽ tính trước các giá trị \(0!, 10^6!, (2 \cdot 10^6)!, (3 \cdot 10^6)!, \ldots, 10^9!\) (tổng cộng \(1001\) số) và lưu vào một mảng \(f\). Việc tính phải được thực hiện bên ngoài, xong rồi bạn chỉ cần dán kết quả vào trong code chính.
\(f\)
1, 641102369, 578095319, 5832229, 259081142, 974067448, 316220877, 690120224, 251368199, 980250487, 682498929, 134623568, 95936601, 933097914, 167332441, 598816162, 336060741, 248744620, 626497524, 288843364, 491101308, 245341950, 565768255, 246899319, 968999, 586350670, 638587686, 881746146, 19426633, 850500036, 76479948, 268124147, 842267748, 886294336, 485348706, 463847391, 544075857, 898187927, 798967520, 82926604, 723816384, 156530778, 721996174, 299085602, 323604647, 172827403, 398699886, 530389102, 294587621, 813805606, 67347853, 497478507, 196447201, 722054885, 228338256, 407719831, 762479457, 746536789, 811667359, 778773518, 27368307, 438371670, 59469516, 5974669, 766196482, 606322308, 86609485, 889750731, 340941507, 371263376, 625544428, 788878910, 808412394, 996952918, 585237443, 1669644, 361786913, 480748381, 595143852, 837229828, 199888908, 526807168, 579691190, 145404005, 459188207, 534491822, 439729802, 840398449, 899297830, 235861787, 888050723, 656116726, 736550105, 440902696, 85990869, 884343068, 56305184, 973478770, 168891766, 804805577, 927880474, 876297919, 934814019, 676405347, 567277637, 112249297, 44930135, 39417871, 47401357, 108819476, 281863274, 60168088, 692636218, 432775082, 14235602, 770511792, 400295761, 697066277, 421835306, 220108638, 661224977, 261799937, 168203998, 802214249, 544064410, 935080803, 583967898, 211768084, 751231582, 972424306, 623534362, 335160196, 243276029, 554749550, 60050552, 797848181, 395891998, 172428290, 159554990, 887420150, 970055531, 250388809, 487998999, 856259313, 82104855, 232253360, 513365505, 244109365, 1559745, 695345956, 261384175, 849009131, 323214113, 747664143, 444090941, 659224434, 80729842, 570033864, 664989237, 827348878, 195888993, 576798521, 457882808, 731551699, 212938473, 509096183, 827544702, 678320208, 677711203, 289752035, 66404266, 555972231, 195290384, 97136305, 349551356, 785113347, 83489485, 66247239, 52167191, 307390891, 547665832, 143066173, 350016754, 917404120, 296269301, 996122673, 23015220, 602139210, 748566338, 187348575, 109838563, 574053420, 105574531, 304173654, 542432219, 34538816, 325636655, 437843114, 630621321, 26853683, 933245637, 616368450, 238971581, 511371690, 557301633, 911398531, 848952161, 958992544, 925152039, 914456118, 724691727, 636817583, 238087006, 946237212, 910291942, 114985663, 492237273, 450387329, 834860913, 763017204, 368925948, 475812562, 740594930, 45060610, 806047532, 464456846, 172115341, 75307702, 116261993, 562519302, 268838846, 173784895, 243624360, 61570384, 481661251, 938269070, 95182730, 91068149, 115435332, 495022305, 136026497, 506496856, 710729672, 113570024, 366384665, 564758715, 270239666, 277118392, 79874094, 702807165, 112390913, 730341625, 103056890, 677948390, 339464594, 167240465, 108312174, 839079953, 479334442, 271788964, 135498044, 277717575, 591048681, 811637561, 353339603, 889410460, 839849206, 192345193, 736265527, 316439118, 217544623, 788132977, 618898635, 183011467, 380858207, 996097969, 898554793, 335353644, 54062950, 611251733, 419363534, 965429853, 160398980, 151319402, 990918946, 607730875, 450718279, 173539388, 648991369, 970937898, 500780548, 780122909, 39052406, 276894233, 460373282, 651081062, 461415770, 358700839, 643638805, 560006119, 668123525, 686692315, 673464765, 957633609, 199866123, 563432246, 841799766, 385330357, 504962686, 954061253, 128487469, 685707545, 299172297, 717975101, 577786541, 318951960, 773206631, 306832604, 204355779, 573592106, 30977140, 450398100, 363172638, 258379324, 472935553, 93940075, 587220627, 776264326, 793270300, 291733496, 522049725, 579995261, 335416359, 142946099, 472012302, 559947225, 332139472, 499377092, 464599136, 164752359, 309058615, 86117128, 580204973, 563781682, 954840109, 624577416, 895609896, 888287558, 836813268, 926036911, 386027524, 184419613, 724205533, 403351886, 715247054, 716986954, 830567832, 383388563, 68409439, 6734065, 189239124, 68322490, 943653305, 405755338, 811056092, 179518046, 825132993, 343807435, 985084650, 868553027, 148528617, 160684257, 882148737, 591915968, 701445829, 529726489, 302177126, 974886682, 241107368, 798830099, 940567523, 11633075, 325334066, 346091869, 115312728, 473718967, 218129285, 878471898, 180002392, 699739374, 917084264, 856859395, 435327356, 808651347, 421623838, 105419548, 59883031, 322487421, 79716267, 715317963, 429277690, 398078032, 316486674, 384843585, 940338439, 937409008, 940524812, 947549662, 833550543, 593524514, 996164327, 987314628, 697611981, 636177449, 274192146, 418537348, 925347821, 952831975, 893732627, 1277567, 358655417, 141866945, 581830879, 987597705, 347046911, 775305697, 125354499, 951540811, 247662371, 343043237, 568392357, 997474832, 209244402, 380480118, 149586983, 392838702, 309134554, 990779998, 263053337, 325362513, 780072518, 551028176, 990826116, 989944961, 155569943, 596737944, 711553356, 268844715, 451373308, 379404150, 462639908, 961812918, 654611901, 382776490, 41815820, 843321396, 675258797, 845583555, 934281721, 741114145, 275105629, 666247477, 325912072, 526131620, 252551589, 432030917, 554917439, 818036959, 754363835, 795190182, 909210595, 278704903, 719566487, 628514947, 424989675, 321685608, 50590510, 832069712, 198768464, 702004730, 99199382, 707469729, 747407118, 302020341, 497196934, 5003231, 726997875, 382617671, 296229203, 183888367, 703397904, 552133875, 732868367, 350095207, 26031303, 863250534, 216665960, 561745549, 352946234, 784139777, 733333339, 503105966, 459878625, 803187381, 16634739, 180898306, 68718097, 985594252, 404206040, 749724532, 97830135, 611751357, 31131935, 662741752, 864326453, 864869025, 167831173, 559214642, 718498895, 91352335, 608823837, 473379392, 385388084, 152267158, 681756977, 46819124, 313132653, 56547945, 442795120, 796616594, 256141983, 152028387, 636578562, 385377759, 553033642, 491415383, 919273670, 996049638, 326686486, 160150665, 141827977, 540818053, 693305776, 593938674, 186576440, 688809790, 565456578, 749296077, 519397500, 551096742, 696628828, 775025061, 370732451, 164246193, 915265013, 457469634, 923043932, 912368644, 777901604, 464118005, 637939935, 956856710, 490676632, 453019482, 462528877, 502297454, 798895521, 100498586, 699767918, 849974789, 811575797, 438952959, 606870929, 907720182, 179111720, 48053248, 508038818, 811944661, 752550134, 401382061, 848924691, 764368449, 34629406, 529840945, 435904287, 26011548, 208184231, 446477394, 206330671, 366033520, 131772368, 185646898, 648711554, 472759660, 523696723, 271198437, 25058942, 859369491, 817928963, 330711333, 724464507, 437605233, 701453022, 626663115, 281230685, 510650790, 596949867, 295726547, 303076380, 465070856, 272814771, 538771609, 48824684, 951279549, 939889684, 564188856, 48527183, 201307702, 484458461, 861754542, 326159309, 181594759, 668422905, 286273596, 965656187, 44135644, 359960756, 936229527, 407934361, 267193060, 456152084, 459116722, 124804049, 262322489, 920251227, 816929577, 483924582, 151834896, 167087470, 490222511, 903466878, 361583925, 368114731, 339383292, 388728584, 218107212, 249153339, 909458706, 322908524, 202649964, 92255682, 573074791, 15570863, 94331513, 744158074, 196345098, 334326205, 9416035, 98349682, 882121662, 769795511, 231988936, 888146074, 137603545, 582627184, 407518072, 919419361, 909433461, 986708498, 310317874, 373745190, 263645931, 256853930, 876379959, 702823274, 147050765, 308186532, 175504139, 180350107, 797736554, 606241871, 384547635, 273712630, 586444655, 682189174, 666493603, 946867127, 819114541, 502371023, 261970285, 825871994, 126925175, 701506133, 314738056, 341779962, 561011609, 815463367, 46765164, 49187570, 188054995, 957939114, 64814326, 933376898, 329837066, 338121343, 765215899, 869630152, 978119194, 632627667, 975266085, 435887178, 282092463, 129621197, 758245605, 827722926, 201339230, 918513230, 322096036, 547838438, 985546115, 852304035, 593090119, 689189630, 555842733, 567033437, 469928208, 212842957, 117842065, 404149413, 155133422, 663307737, 208761293, 206282795, 717946122, 488906585, 414236650, 280700600, 962670136, 534279149, 214569244, 375297772, 811053196, 922377372, 289594327, 219932130, 211487466, 701050258, 398782410, 863002719, 27236531, 217598709, 375472836, 810551911, 178598958, 247844667, 676526196, 812283640, 863066876, 857241854, 113917835, 624148346, 726089763, 564827277, 826300950, 478982047, 439411911, 454039189, 633292726, 48562889, 802100365, 671734977, 945204804, 508831870, 398781902, 897162044, 644050694, 892168027, 828883117, 277714559, 713448377, 624500515, 590098114, 808691930, 514359662, 895205045, 715264908, 628829100, 484492064, 919717789, 513196123, 748510389, 403652653, 574455974, 77123823, 172096141, 819801784, 581418893, 15655126, 15391652, 875641535, 203191898, 264582598, 880691101, 907800444, 986598821, 340030191, 264688936, 369832433, 785804644, 842065079, 423951674, 663560047, 696623384, 496709826, 161960209, 331910086, 541120825, 951524114, 841656666, 162683802, 629786193, 190395535, 269571439, 832671304, 76770272, 341080135, 421943723, 494210290, 751040886, 317076664, 672850561, 72482816, 493689107, 135625240, 100228913, 684748812, 639655136, 906233141, 929893103, 277813439, 814362881, 562608724, 406024012, 885537778, 10065330, 60625018, 983737173, 60517502, 551060742, 804930491, 823845496, 727416538, 946421040, 678171399, 842203531, 175638827, 894247956, 538609927, 885362182, 946464959, 116667533, 749816133, 241427979, 871117927, 281804989, 163928347, 563796647, 640266394, 774625892, 59342705, 256473217, 674115061, 918860977, 322633051, 753513874, 393556719, 304644842, 767372800, 161362528, 754787150, 627655552, 677395736, 799289297, 846650652, 816701166, 687265514, 787113234, 358757251, 701220427, 607715125, 245795606, 600624983, 10475577, 728620948, 759404319, 36292292, 491466901, 22556579, 114495791, 647630109, 586445753, 482254337, 718623833, 763514207, 66547751, 953634340, 351472920, 308474522, 494166907, 634359666, 172114298, 865440961, 364380585, 921648059, 965683742, 260466949, 117483873, 962540888, 237120480, 620531822, 193781724, 213092254, 107141741, 602742426, 793307102, 756154604, 236455213, 362928234, 14162538, 753042874, 778983779, 25977209, 49389215, 698308420, 859637374, 49031023, 713258160, 737331920, 923333660, 804861409, 83868974, 682873215, 217298111, 883278906, 176966527, 954913, 105359006, 390019735, 10430738, 706334445, 315103615, 567473423, 708233401, 48160594, 946149627, 346966053, 281329488, 462880311, 31503476, 185438078, 965785236, 992656683, 916291845, 881482632, 899946391, 321900901, 512634493, 303338827, 121000338, 967284733, 492741665, 152233223, 165393390, 680128316, 917041303, 532702135, 741626808, 496442755, 536841269, 131384366, 377329025, 301196854, 859917803, 676511002, 373451745, 847645126, 823495900, 576368335, 73146164, 954958912, 847549272, 241289571, 646654592, 216046746, 205951465, 3258987, 780882948, 822439091, 598245292, 869544707, 698611116
Khi đó, để tìm \(n!\), ta chỉ cần tìm \(f_i\) gần với \(n!\) nhất rồi tính bằng công thức \(n! = n \cdot (n - 1) \cdot (n - 2) \cdot \ldots \cdot f_i\).
Vì khoảng cách giữa hai \(f\) liên tiếp chỉ có \(10^6\) nên số lần duyệt để tính sẽ không quá \(10^6\).
Độ phức tạp: \(O(10^6)\).
Solution
C++
C++
#include <bits/stdc++.h>
using namespace std;
const int mod = 1e9 + 7;
int f[] = {}; // 1001 numbers 0e6!, 1e6!, 2e6!, ..., 1e9!
int factorial(int n) {
int res = f[n / int(1e6)];
for (int i = n / int(1e6) * int(1e6) + 1; i <= n; i++) {
res = res * (long long)i % mod;
}
return res;
}
int inverse(int x) {
if (x <= 1) {
return 1;
}
return (mod - mod / x) * (long long)inverse(mod % x) % mod;
}
int choose(int n, int k) {
if (n < 0 || k < 0 || n < k) {
return 0;
}
return factorial(n) * (long long)inverse(factorial(n - k) * (long long)factorial(k) % mod) % mod;
}
int main() {
int n, k;
cin >> n >> k;
cout << choose(n, k) << "\n";
return 0;
}
Pascal
Delphi
const
modulo = 1000000007;
var
f: array[0..1000] of longint = (); // 1001 numbers 0e6!, 1e6!, 2e6!, ..., 1e9!
function factorial(n: longint): int64;
var
i: longint;
res: int64;
begin
res := f[n div 1000000];
for i := n div 1000000 * 1000000 + 1 to n do
begin
res := res * i mod modulo;
end;
exit(res);
end;
function inverse(x: longint): int64;
begin
if x <= 1 then
begin
exit(1);
end;
exit((modulo - modulo div x) * inverse(modulo mod x) mod modulo);
end;
function choose(n, k: longint): int64;
begin
if (n < 0) or (k < 0) or (n < k) then
begin
exit(0);
end;
exit(factorial(n) * inverse(factorial(n - k) * factorial(k) mod modulo) mod modulo);
end;
var
n, k: longint;
begin
readln(n, k);
writeln(choose(n, k));
end.
Python
Python
mod = 10**9 + 7
f = [] # 1001 numbers 0e6!, 1e6!, 2e6!, ..., 1e9!
def factorial(n):
res = f[n // 10**6]
for i in range(n // 10**6 * 10**6 + 1, n + 1):
res = res * i % mod
return res
def inverse(x):
if x <= 1:
return 1
return (mod - mod // x) * inverse(mod % x) % mod
def choose(n, k):
if n < 0 or k < 0 or n < k:
return 0
return factorial(n) * inverse(factorial(n - k) * factorial(k) % mod) % mod
n, k = map(int, input().split())
print(choose(n, k))
Subtask \(5\):
Tutorial
Sủ dụng định lý Lucas.
Độ phức tạp: \(O(10^6)\).
Bình luận
chi tiết zậy:))
Đọc xong cái đầu muốn nổ tung luôn.