CF GYM100548 (number of schemes with different adjacent lattice colors) 2014 XI ' an regional race F-Question tolerance principle

n Grid row, there are M color, ask the exact k color to dye, so that the adjacent lattice color of the scheme number.

Integers n, m, K (1≤n, M≤10^9, 1≤k≤10^6, K≤n, M).

M color take K species C (M, k) This can be put in the last multiply then the problem becomes only k color
The first lattice has K-type coating method The second one has k-1 species, the third is k-1 species.

Altogether is k* (k-1) ^ (n-1) This algorithm only guarantees the adjacent color is different, the total color number does not exceed the K kind of, and does not guarantee exactly to appear the K color namely many forget exactly appears 2 kinds of exactly appear 3 kinds .... Happen to appear k-1 species

We were asking for the right kind of k, and now we're asking for k-1.
So that's (k-1) * (k-2) ^ (n-1) and this is a lot of things.
And so on, then you can use the repulsion principle.

For example, there are 5 colors, choose 4 kinds is

C (5, 4) * (C (4, 4) *4*3^4-c (4, 3) *3*2^4 + C (4, 2) *2*1^4)

Sample Input

2
3 2 2//N m k
3 2 1
Sample Output

Case #1:2
Case #2:0

1# include <iostream>2# include <cstdio>3# include <cstring>4# include <algorithm>5# include <string>6# include <cmath>7# include <queue>8# include <list>9# define LLLong LongTen using namespacestd; One  A Const intMOD =1000000007 ; -  - intN, M, K; the LL CM; -LL ck[1000010] ; -LL inv[1000010] ; -  +  - ll Pow_mod (ll P, ll K) + { ALL ans =1; at      while(k) { -         if(K &1) ans = ans * p%MOD; -p = (LL) p*p%MOD; -K >>=1; -     } -     returnans; in } -  toll EXT_GCD (ll a,ll b,ll &x,ll &y) {//Extended Euclid +    if(a==0&&b==0)return-1; -    if(b==0) {x=1, y=0;returnA;} theLL d= EXT_GCD (b,a%b,y,x); *y-= a/b*x; $    returnD;Panax Notoginseng } - //ax = 1 (mod m) thell Inv (ll a,ll m) {//finding inverse of inverse element A to M +LL d,x,y,t =m; AD=EXT_GCD (a,t,x,y); the    if(d==1)return(x%t+t)%T; +    return-1; - } $  $  -ll Cm (ll N, ll M, ll P)//To find the number of combinations - { theLL a=1, b=1; -     if(m>n)return 0;Wuyi      while(m) the     { -A= (a*n)%p; Wub= (b*m)%p; -m--; Aboutn--; $     } -     return(LL) A*INV (b,p)%p;//(A/b)%p equivalent to A * (b,p) Inverse - } -  A intLucas (ll N, ll M, ll P)//multiply the N-piecewise recursive solution + { the     if(m==0)return 1; -     return(LL) Cm (n%p,m%p,p) * (LL) Lucas (n/p,m/p,p)%p; $ } the  the voidInit () the { theinv[1] =1 ; -     inti; in      for(i =2; I <1000010; i++) theInv[i] =INV (i,mod); the } About  the intMain () the { the     //freopen ("In.txt", "R", stdin); +     intT; -scanf"%d", &T); the     intCase =0 ;Bayi init (); the      while(t--) the     { -case++ ; -scanf"%d%d%d", &n, &m, &k); the         if(n = =1) the         { theprintf"Case #%d:%d\n", case, m); the             Continue ; -         } the         inti; theCM =Cm (m,k,mod); theck[0] =1 ;94          for(i =1; I <= K; i++) theCk[i] = (ck[i-1] * (k-i+1)%mod * Inv[i])%MOD; theLL ans =0, t =1 ; the          for(i = k; I >=2; i--)98         { AboutAns = (ans + t*ck[i]*i%mod*pow_mod (i-1, N-1)%mod+mod)%MOD; -T *=-1 ;101         }102printf"Case #%d:%i64d\n", case,ans*cm%MOD);103 104     } the     return 0;106}

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CF GYM100548 (number of schemes with different adjacent lattice colors) 2014 XI ' an regional race F-question tolerance principle)