class Modulo_Polynominal(): def __init__(self,Poly,Max_Degree=2*10**5,Char="X"): from itertools import zip_longest """多項式の定義 P:係数のリスト C:文字 Max_Degree ※Mod:法はグローバル変数から指定 """ self.Poly=[p%Mod for p in Poly][:Max_Degree] self.Char=Char self.Max_Degree=Max_Degree def __str__(self): S="" flag=False for k in range(len(self.Poly)): if self.Poly[k]: if flag: if k==1: S+="{:+}{}".format(self.Poly[1],self.Char) else: S+="{:+}{}^{}".format(self.Poly[k],self.Char,k) else: flag=True if k==0: S=str(self.Poly[0]) elif k==1: S=str(self.Poly[1])+self.Char else: S=str(self.Poly[k])+"{}^{}".format(self.Char,k) if S: return S else: return "0" #+,- def __pos__(self): return self def __neg__(self): return self.scale(-1) #Boole def __bool__(self): for a in self.Poly: if a: return True return False #簡略化 def reduce(self): P_deg=self.degree() if not(P_deg>=0): T=Modulo_Polynominal([0],self.Max_Degree,self.Char) T.censor(self.Max_Degree) return T for i in range(self.degree(),-1,-1): if self.Poly[i]: T=Modulo_Polynominal(self.Poly[:i+1],self.Max_Degree,self.Char) T.censor(self.Max_Degree) return T #次数 def degree(self): x=-float("inf") k=0 for y in self.Poly: if y!=0: x=k k+=1 return x #加法 def __add__(self,other): P=self Q=other if Q.__class__==Modulo_Polynominal: from itertools import zip_longest N=min(P.Max_Degree,Q.Max_Degree) R=[(a+b)%Mod for (a,b) in zip_longest(P.Poly,Q.Poly,fillvalue=0)] return Modulo_Polynominal(R,N,P.Char) else: P_deg=P.degree() R=[0]*(P_deg+1) R=[p for p in P.Poly] R[0]=(R[0]+Q)%Mod return Modulo_Polynominal(R,P.Max_Degree,P.Char).reduce() def __radd__(self,other): return self+other #減法 def __sub__(self,other): return self+(-other) def __rsub__(self,other): return (-self)+other #乗法 def __mul__(self,other): P=self Q=other if Q.__class__==Modulo_Polynominal: M=min(P.Max_Degree,Q.Max_Degree) B=Convolution_Mod(self.Poly,other.Poly)[:M] return Modulo_Polynominal(B,M,self.Char).reduce() else: return self.scale(other) def __rmul__(self,other): return self.scale(other) #除法 def __floordiv__(self,other): if not other: raise ZeroDivisionError pass #剰余 def __mod__(self,other): return self-(self//other)*other #累乗 def __pow__(self,n): m=abs(n) Q=self A=Modulo_Polynominal([1],self.Max_Degree,self.Char) while m>0: if m&1: A*=Q m>>=1 Q*=Q if n>=0: return A else: return A.__inv__() #逆元 def __inv__(self,deg=None): assert self.Poly[0],"定数項が0" P=self if deg==None: deg=P.Max_Degree else: deg=min(deg,P.Max_Degree) F=P.Poly N=len(F) r=pow(F[0],Mod-2,Mod) m=1 T=[r] while mn: E=P.Poly[:n] else: E=P.Poly+[0]*(n-P.Poly) return Modulo_Polynominal(E,P.Max_Degree,P.Char) else: if len(P.Poly)>n: del P.Poly[n:] else: P.Poly+=[0]*(n-len(P.Poly)) #================================================= def Primitive_Root(p): """Z/pZ上の原始根を見つける p:素数 """ if p==2: return 1 if p==998244353: return 3 if p==10**9+7: return 5 if p==163577857: return 23 if p==167772161: return 3 if p==469762049: return 3 fac=[] q=2 v=p-1 while v>=q*q: e=0 while v%q==0: e+=1 v//=q if e>0: fac.append(q) q+=1 if v>1: fac.append(v) g=2 while g>e S=[pow(primitive,(Mod-1)>>i,Mod) for i in range(e+1)] for l in range(H, 0, -1): d = 1 << l - 1 U = [1]*(d+1) u = 1 for i in range(d): u=u*S[l]%Mod U[i+1]=u for i in range(1 <>e inv_primitive=pow(primitive,Mod-2,Mod) S=[pow(inv_primitive,(Mod-1)>>i,Mod) for i in range(e+1)] for l in range(1, H + 1): d = 1 << l - 1 for i in range(1 << H - l): u = 1 for j in range(i * 2 * d, (i * 2 + 1) * d): A[j+d] *= u A[j], A[j+d] = (A[j] + A[j+d]) % Mod, (A[j] - A[j+d]) % Mod u = u * S[l] % Mod N_inv=pow(N,Mod-2,Mod) for i in range(N): A[i]=A[i]*N_inv%Mod #参考元 https://atcoder.jp/contests/practice2/submissions/16789717 def Convolution_Mod(A,B): """A,BをMod を法とする畳み込みを求める. ※Modはグローバル変数から指定 """ if not A or not B: return [] N=len(A) M=len(B) L=N+M-1 if min(N,M)<=50: if NP.Max_Degree: return Modulo_Polynominal([0],P.Max_Degree,P.Char) p_inv=pow(p,Mod-2,Mod) Q=Modulo_Polynominal([(p_inv*a)%Mod for a in F[d:]],P.Max_Degree,P.Char) G=Exp(k*Log(Q)).Poly pk=pow(p,k,Mod) G=[0]*(d*k)+[(pk*a)%Mod for a in G] return Modulo_Polynominal(G,P.Max_Degree,P.Char) #ルジャンドル記号 def Legendre(X): """ルジャンドル記号(a/p)を返す. ※法が素数のときのみ成立する. """ if X==0: return 0 elif pow(X,(Mod-1)//2,Mod)==1: return 1 else: return -1 #根号 def Tonelli_Shanks(X): """X=a (mod p)のとき,r*r=a (mod p)を満たすrを返す. ※法pが素数のときのみ有効 ※存在しないときはNoneが返り値 """ X%=Mod if Legendre(X)==-1: return None from random import randint as ri if X==0: return X elif Mod==2: return X elif Mod%4==3: return pow(X,(Mod+1)//4,Mod) u=2 s=1 while (Mod-1)%(2*u)==0: u*=2 s+=1 q=(Mod-1)//u z=0 while pow(z,(Mod-1)//2,Mod)!=Mod-1: z=ri(1,Mod-1) m,c,t,r=s,pow(z,q,Mod),pow(X,q,Mod),pow(X,(q+1)//2,Mod) while m>1: if pow(t,2**(m-2),Mod)==1: c=(c*c)%Mod m=m-1 else: c,t,r,m=(c*c)%Mod,(c*c*t)%Mod,(c*r)%Mod,m-1 return r #多項式の根号 def __sqrt(F,N): F+=[0]*(N-len(F)) s=Tonelli_Shanks(F[0]) if s==None:return None m=1 G=[min(s,Mod-s)] two_inv=pow(2,Mod-2,Mod) while m0: E=[0]*(d//2)+E return Modulo_Polynominal(E,P.Max_Degree,P.Char) #Bernoulli def Bernoulli(N): """ ベルヌーイ数 B_0,B_1,...,B_Nの(mod Mod)での値を求める. """ if N==0: return [1] X=Modulo_Polynominal([0,1],N+2) P=Exp(X) del P.Poly[0] F=(1/P).Poly fact=1 for i in range(N+1): F[i]=(F[i]*fact)%Mod fact=(fact*(i+1))%Mod del F[-1] return F def Subset_Sum(X,K): """Xの要素のうち,任意個を用いて, 和がk=0,1,...,Kになる組み合わせの総数をModで割った余りを求める. X:リスト (X[0]=0) K:非負整数 """ A=[0]*(K+1) for x in X: if x<=K: A[x]+=1 Inv=[0]*(K+1) Inv[1]=1 for i in range(2,K+1): Inv[i]=(-(Mod//i)*Inv[Mod%i])%Mod F=[0]*(K+1) for i in range(1,K+1): j=i k=1 c=1 while j<=K: F[j]=(F[j]+c*Inv[k]*A[i])%Mod c*=-1 j+=i k+=1 P=Modulo_Polynominal(F,K+1) return Exp(P).Poly class Modulo_Error(Exception): pass class Modulo(): def __init__(self,a,n): self.a=a%n self.n=n def __str__(self): return "{} (mod {})".format(self.a,self.n) #+,- def __pos__(self): return self def __neg__(self): return Modulo(-self.a,self.n) #等号,不等号 def __eq__(self,other): if isinstance(other,Modulo): return (self.a==other.a) and (self.n==other.n) elif isinstance(other,int): return (self-other).a==0 def __neq__(self,other): return not(self==other) #加法 def __add__(self,other): if isinstance(other,Modulo): if self.n!=other.n: raise Modulo_Error("異なる法同士の演算です.") return Modulo(self.a+other.a,self.n) elif isinstance(other,int): return Modulo(self.a+other,self.n) def __radd__(self,other): if isinstance(other,int): return Modulo(self.a+other,self.n) #減法 def __sub__(self,other): return self+(-other) def __rsub__(self,other): if isinstance(other,int): return -self+other #乗法 def __mul__(self,other): if isinstance(other,Modulo): if self.n!=other.n: raise Modulo_Error("異なる法同士の演算です.") return Modulo(self.a*other.a,self.n) elif isinstance(other,int): return Modulo(self.a*other,self.n) def __rmul__(self,other): if isinstance(other,int): return Modulo(self.a*other,self.n) #Modulo逆数 def inverse(self): return self.Modulo_Inverse() def Modulo_Inverse(self): x0, y0, x1, y1 = 1, 0, 0, 1 a,b=self.a,self.n while b != 0: q, a, b = a // b, b, a % b x0, x1 = x1, x0 - q * x1 y0, y1 = y1, y0 - q * y1 if a!=1: raise Modulo_Error("{}の逆数が存在しません".format(self)) else: return Modulo(x0,self.n) #除法 def __truediv__(self,other): return self*(other.Modulo_Inverse()) def __rtruediv__(self,other): return other*(self.Modulo_Inverse()) #累乗 def __pow__(self,m): u=abs(m) r=Modulo(pow(self.a,u,self.n),self.n) if m>=0: return r else: return r.Modulo_Inverse() #法の合成 def __modulo_composite__(p,q): from math import gcd a,n=p.a,p.n b,m=q.a,q.n d=b-a g,h=n,m while h: g,h=h,g%h if d%g: raise Modulo_Error("{}と{}は両立しません.".format(p,q)) n//=g m//=g d//=g s=(Modulo(1,m)/Modulo(n,m)).a return Modulo(a+(n*g)*d*s,n*m*g) def Modulo_Composite(*X): from functools import reduce return reduce(__modulo_composite__,X) #================================================= alpha=167772161 beta =469762049 gamma=1224736769 K=int(input()) N=int(input()) X=list(map(int,input().split())) L=[0]*(K+1) for x in X: if x<=K: L[x]+=1 A=[] for Mod in [alpha,beta,gamma]: P=Modulo_Polynominal(L,K+1) Q=1/(1-P) A.append(Modulo(Q.coefficient(K),Mod)) B=Modulo_Composite(*A).a print(B%(10**9+7))