You are given an array $\mbox{A}$ of size $N$. You are asked to answer $\mbox{Q}$ queries.  

Each query contains a number $\mbox{M}$.  

For each distinct permutation of the array $\mbox{A}$, you need to print the sum of the values returned by the find function.

As the sum can be too large, output it modulo $\mbox{P}$, which is a prime and given in input.  

find(int permutation_A[], int M)
{
    x = Length(permutation_A)
    sum = 0
    for(i = 0; i < x; i++) {
        if (permutation_A[i] <= M)
            sum = sum + permutation_A[i]
        else
            break
    }
    return sum
}

Input Format 

The first line of input contains $\mbox{P}$. 

The second line of input contains $N$. The next line contains $N$ numbers $A[0]\ldots A[N-1]$ separated by single spaces. 

The next line contains $\mbox{Q}$, the number of queries to follow. Each subsequent line contains one positive integer $\mbox{M}$.  

Output Format 

For each query, output as asked above.  

Constraints 

$1000000\leq P\leq2000003$ 

$1\leq N\leq10^5$ 

$1\leq Q\leq10^5$ 

$1\leq A[i],M\leq10^9$

Sample Input

2000003
5
4 2 3 1 4
2
1
2

Sample Output

12
45

Explanation

Query 1: 

Consider all permutations. if the first number is greater than 1, then the loop will break in the beginning itself. There are a total of 60 distinct permutations out of which 48 will give 0. The remaining 12 will fetch 1 each from the function. Thus the answer is 12.

Difficulty: MEDIUM_HARD
Tags: Combinatorics

Starter code:

Solutions:
from collections import defaultdict
from bisect import bisect_left, bisect_right
from itertools import product
mod = int(input().strip())

def binomial(n, k):
	if n < k or k < 0:
		return 0
	return fac[n] * invfac[k] * invfac[n - k] % mod

def sum_(n, k):
	return (n * binomial(n + 1, k + 1) - binomial(n + 1, k + 2)) % mod

def init(counter):
	n = len(counter)
	A = sorted(counter.keys())
	C = [counter[x] for x in A]
	N = sum(C)
	T = 0
	A_C = 0
	S = [0] * (n + 1)
	p = 1
	for c in C:
		p = p * invfac[c] % mod
	S[n] = p
	for i in range(1, n):
		T += C[i - 1]
		A_C += A[i - 1] * C[i - 1]
		S[i] = fac[N - T] * fac[T - 1] % mod
		S[i] = S[i] * A_C * p % mod
		S[i] = S[i] * ((N - 1) * binomial(N - 1, T - 1) - sum_(N - 2, N - 1 - T)) % mod
	A_C += A[n - 1] * C[n - 1]
	T += C[n - 1]
	S[n] = S[n] * fac[T] * A_C % mod
	return (S, A)

def brute_force(counter, M):
	A = sorted(counter.keys())
	C = [counter[x] for x in A]
	T = 0
	for i in range(len(A)):
		if A[i] <= M:
			T += C[i]
		else:
			break
	N = sum(C)
	ans = 0
	for k in range(1, T + 1):
		loc_ans = 0
		for t in product(*[range(C[j] + 1) for j in range(i)]):
			s = 1
			if sum(t) != k:
				continue
			for j in range(i):
				s = s * binomial(C[j], t[j])
			s = s * sum((t[j] * A[j] for j in range(i))) % mod
			loc_ans = (loc_ans + s) % mod
		ans = (ans + loc_ans * fac[N - k - 1] * fac[k]) % mod
	ans = ans * (N - T) % mod
	for j in range(len(C)):
		ans = ans * invfac[C[j]] % mod
	return ans
N_max = 10 ** 5
fac = [1] * (N_max + 1)
invfac = [1] * (N_max + 1)
for i in range(2, N_max + 1):
	fac[i] = fac[i - 1] * i % mod
invfac[N_max] = pow(fac[N_max], mod - 2, mod)
for i in reversed(range(2, N_max)):
	invfac[i] = invfac[i + 1] * (i + 1) % mod
N = int(input().strip())
A = list(map(int, input().strip().split()))
counter = defaultdict(int)
for x in A:
	counter[x] += 1
(S, A) = init(counter)
Q = int(input().strip())
for _ in range(Q):
	M = int(input().strip())
	i = bisect_right(A, M)
	print(S[i])


p = int(input())
n = int(input())
a = sorted(list(map(int, input().split())))
suma = [0]
todiv = 1
currval = -1
currreps = 1
thefact = 1
for i in range(n):
	thefact = thefact * (i + 1) % p
	x = a[i]
	suma.append(suma[-1] + x)
	if x == currval:
		currreps += 1
		todiv = todiv * currreps % p
	else:
		currval = x
		currreps = 1
mult = pow(todiv, p - 2, p)
q = int(input())
for _ in range(q):
	query = int(input())
	left = -1
	right = len(a)
	while right - left > 1:
		mid = (left + right) // 2
		if a[mid] <= query:
			left = mid
		else:
			right = mid
	print(thefact * suma[right] * pow(n - left, p - 2, p) * mult % p)


from collections import defaultdict
from bisect import bisect_left, bisect_right
from itertools import product
mod = int(input().strip())

def binomial(n, k):
	if n < k or k < 0:
		return 0
	return fac[n] * invfac[k] * invfac[n - k] % mod

def sum_(n, k):
	return (n * binomial(n + 1, k + 1) - binomial(n + 1, k + 2)) % mod

def init(counter):
	n = len(counter)
	A = sorted(counter.keys())
	C = [counter[x] for x in A]
	N = sum(C)
	T = 0
	A_C = 0
	S = [0] * (n + 1)
	p = 1
	for c in C:
		p = p * invfac[c] % mod
	S[n] = p
	for i in range(1, n):
		T += C[i - 1]
		A_C += A[i - 1] * C[i - 1]
		S[i] = fac[N - T] * fac[T - 1] % mod
		S[i] = S[i] * A_C * p % mod
		S[i] = S[i] * ((N - 1) * binomial(N - 1, T - 1) - sum_(N - 2, N - 1 - T)) % mod
	A_C += A[n - 1] * C[n - 1]
	T += C[n - 1]
	S[n] = S[n] * fac[T] * A_C % mod
	return (S, A)

def brute_force(counter, M):
	A = sorted(counter.keys())
	C = [counter[x] for x in A]
	T = 0
	for i in range(len(A)):
		if A[i] <= M:
			T += C[i]
		else:
			break
	N = sum(C)
	ans = 0
	for k in range(1, T + 1):
		loc_ans = 0
		for t in product(*[range(C[j] + 1) for j in range(i)]):
			s = 1
			if sum(t) != k:
				continue
			for j in range(i):
				s = s * binomial(C[j], t[j])
			s = s * sum((t[j] * A[j] for j in range(i))) % mod
			loc_ans = (loc_ans + s) % mod
		ans = (ans + loc_ans * fac[N - k - 1] * fac[k]) % mod
	ans = ans * (N - T) % mod
	for j in range(len(C)):
		ans = ans * invfac[C[j]] % mod
	return ans
N_max = 10 ** 5
fac = [1] * (N_max + 1)
invfac = [1] * (N_max + 1)
for i in range(2, N_max + 1):
	fac[i] = fac[i - 1] * i % mod
invfac[N_max] = pow(fac[N_max], mod - 2, mod)
for i in reversed(range(2, N_max)):
	invfac[i] = invfac[i + 1] * (i + 1) % mod
N = int(input().strip())
A = list(map(int, input().strip().split()))
counter = defaultdict(int)
for x in A:
	counter[x] += 1
(S, A) = init(counter)
Q = int(input().strip())
for _ in range(Q):
	M = int(input().strip())
	i = bisect_right(A, M)
	print(S[i])