Problem:


Score : 400 points

Problem Statement

We have a large square grid with H rows and W columns.
Iroha is now standing in the top-left cell.
She will repeat going right or down to the adjacent cell, until she reaches the bottom-right cell.

However, she cannot enter the cells in the intersection of the bottom A rows and the leftmost B columns. (That is, there are A×B forbidden cells.) There is no restriction on entering the other cells.

Find the number of ways she can travel to the bottom-right cell.

Since this number can be extremely large, print the number modulo 10^9+7.

Constraints

 1 ≦ H, W ≦ 100,000

 1 ≦ A < H

 1 ≦ B < W

Input

The input is given from Standard Input in the following format:

H W A B

Output

Print the number of ways she can travel to the bottom-right cell, modulo 10^9+7.

Sample Input 1

2 3 1 1

Sample Output 1

2

We have a 2×3 grid, but entering the bottom-left cell is forbidden. The number of ways to travel is two: "Right, Right, Down" and "Right, Down, Right".

Sample Input 2

10 7 3 4

Sample Output 2

3570

There are 12 forbidden cells.

Sample Input 3

100000 100000 99999 99999

Sample Output 3

1

Sample Input 4

100000 100000 44444 55555

Sample Output 4

738162020



Sample input:
2 3 1 1


Sample output:
2

Submission status: Accepted

Solution code:
import sys
import numpy as np
from heapq import heappush, heappop 
from bisect import bisect_left as bi_l, bisect_right as bi_r
from collections import deque, Counter, defaultdict
from itertools import combinations, product
import string 
inf = float('inf')
MOD = 10**9+7
# MOD = 998244353


class NumberTheory():
  def __init__(self, n=2*10**6, numpy=True):
    self.n = n
    self.np_flg = numpy 
    self.is_prime_number, self.prime_numbers = self.sieve_of_eratosthenes(n)
  
  def sieve_of_eratosthenes(self, n):
    if self.np_flg:
      sieve = np.ones(n+1, dtype=np.int64); sieve[:2] = 0
      for i in range(2, int(n**.5)+1):
        if sieve[i]: sieve[i*2::i] = 0
      prime_numbers = np.flatnonzero(sieve)
    else:
      sieve = [1] * (n+1); sieve[0] = sieve[1] = 0
      for i in range(2, int(n**.5)+1):
        if not sieve[i]: continue
        for j in range(i*2, n+1, i): sieve[j] = 0
      prime_numbers = [i for i in range(2, n+1) if sieve[i]]
    return sieve, prime_numbers 

  def prime_factorize(self, n):
    res = dict()
    if n < 2: return res
    border = int(n**.5)
    for p in self.prime_numbers:
      if p > border: break
      while n % p == 0: res[p] = res.get(p, 0)+1; n //= p
      if n == 1: return res
    res[n] = 1; return res

  def prime_factorize_factorial(self, n):
    res = dict()
    for i in range(2, n+1):
      for p, c in self.prime_factorize(i).items(): res[p] = res.get(p, 0)+c
    return res
  
  @staticmethod
  def gcd(a, b): return gcd(b, a%b) if b else abs(a)

  @staticmethod
  def lcm(a, b): return abs(a // gcd(a, b) * b)

  @staticmethod
  def find_divisors(n):
    divisors = []
    for i in range(1, int(n**.5)+1):
      if n%i: continue
      divisors.append(i)
      j = n // i
      if j != i: divisors.append(j)
    return divisors
  
  @staticmethod
  def base_convert(n, b):
    if not n: return [0]
    res = []
    while n:
      n, r = divmod(n, b)
      if r < 0: n += 1; r -= b
      res.append(r)
    return res


class UnionFind():
  def __init__(self, n=10**6):
    self.root = list(range(n))
    self.height = [0] * n 
    self.size = [1] * n 
  
  def find_root(self, u):
    if self.root[u] == u: return u
    self.root[u] = self.find_root(self.root[u])
    return self.root[u]
  
  def unite(self, u, v):
    ru = self.find_root(u)
    rv = self.find_root(v)
    if ru == rv: return 
    hu = self.height[ru]
    hv = self.height[rv]
    if hu >= hv:
      self.root[rv] = ru 
      self.size[ru] += self.size[rv]
      self.height[ru] = max(hu, hv+1)
    else:
      self.root[ru] = rv 
      self.size[rv] += self.size[ru]


class Combinatorics():
  def __init__(self, N=10**9, n=10**6, mod=10**9+7, numpy=True):
    self.mod = mod
    self.nCr = dict()
    self.np_flg=numpy
    self.make_mod_tables(N, n)

  sys.setrecursionlimit(10**6)
  def choose(self, n, r, mod=None): # no mod, or mod ≠ prime
    if r > n or r < 0: return 0
    if r == 0: return 1
    if (n, r) in self.nCr: return self.nCr[(n, r)]
    if not mod:
      self.nCr[(n, r)] = (self.choose(n-1, r) + self.choose(n-1, r-1))
    else:
      self.nCr[(n, r)] = (self.choose(n-1, r, mod) + self.choose(n-1, r-1, mod)) % mod
    return self.nCr[(n,r)]
  
  def cumprod(self, a):
    p = self.mod
    l = len(a); sql = int(np.sqrt(l)+1)
    a = np.resize(a, sql**2).reshape(sql, sql)
    for i in range(sql-1): a[:, i+1] *= a[:, i]; a[:, i+1] %= p
    for i in range(sql-1): a[i+1] *= a[i, -1]; a[i+1] %= p
    return np.ravel(a)[:l]

  def make_mod_tables(self, N, n):
    p = self.mod
    if self.np_flg:
      fac = np.arange(n+1); fac[0] = 1; fac = self.cumprod(fac)
      ifac = np.arange(n+1, 0, -1); ifac[0] = pow(int(fac[-1]), p-2, p)
      ifac = self.cumprod(ifac)[n::-1]
      n_choose = np.arange(N+1, N-n, -1); n_choose[0] = 1;
      n_choose[1:] = self.cumprod(n_choose[1:])*ifac[1:n+1]%p
    else:
      fac = [None]*(n+1); fac[0] = 1
      for i in range(n): fac[i+1] = fac[i]*(i+1)%p
      ifac = [None]*(n+1); ifac[n] = pow(fac[n], p-2, p)
      for i in range(n, 0, -1): ifac[i-1] = ifac[i]*i%p
      n_choose = [None] * (n+1); n_choose[0] = 1
      for i in range(n): n_choose[i+1] = n_choose[i]*(N-i)%p
      for i in range(n+1): n_choose[i] = n_choose[i]*ifac[i]%p
    self.fac, self.ifac, self.mod_n_choose = fac, ifac, n_choose
  
  def mod_choose(self, n, r):
    return self.fac[n]*self.ifac[r]%self.mod*self.ifac[n-r]%self.mod


def z_algorithm(s):
  n = len(s)
  a = [0] * n; a[0] = n
  l = r = -1
  for i in range(1, n):
    if r >= i: a[i] = min(a[i-l], r-i)
    while i + a[i] < n and s[i+a[i]] == s[a[i]]: a[i] += 1
    if i+a[i] >= r: l, r = i, i+a[i]
  return a


class ABC001():
  def A():
    h1, h2 = map(int, sys.stdin.read().split())
    print(h1-h2)

  def B(): pass
  def C(): pass 
  def D(): pass 


class ABC002():
  def A():
    x, y = map(int, sys.stdin.readline().split())
    print(max(x, y))

  def B():
    vowels = set('aeiou')
    s = sys.stdin.readline().rstrip() 
    t = ''
    for c in s:
      if c in vowels: continue 
      t += c 
    print(t)

  def C():
    *coords, = map(int, sys.stdin.readline().split())
    def triangle_area(x0, y0, x1, y1, x2, y2):
      x1 -= x0; x2 -= x0; y1 -= y0; y2 -= y0;
      return abs(x1*y2 - x2*y1) / 2
    print(triangle_area(*coords))

  def D():
    n, m = map(int, sys.stdin.readline().split())
    edges = set()
    for _ in range(m):
      x, y = map(int, sys.stdin.readline().split())
      x -= 1; y -= 1
      edges.add((x, y))
    cand = []
    for i in range(1, 1<<n):
      s = [j for j in range(n) if i>>j & 1]
      for x, y in combinations(s, 2):
        if (x, y) not in edges: break
      else:
        cand.append(len(s))
    print(max(cand))


class ABC003():
  def A():
    n = int(sys.stdin.readline().rstrip())
    print((n+1)*5000)

  def B():
    atcoder = set('atcoder')
    s, t = sys.stdin.read().split()
    for i in range(len(s)):
      if s[i] == t[i]: continue
      if s[i] == '@' and t[i] in atcoder: continue 
      if t[i] == '@' and s[i] in atcoder: continue
      print('You will lose')
      return 
    print('You can win')

  def C():
    n, k, *r = map(int, sys.stdin.read().split())
    res = 0
    for x in sorted(r)[-k:]:
      res = (res+x) / 2
    print(res)

  def D(): pass 


class ABC004():
  def A():
    print(int(sys.stdin.readline().rstrip())*2)

  def B():
    c = [sys.stdin.readline().rstrip() for _ in range(4)]
    for l in c[::-1]:
      print(l[::-1])

  def C():
    n = int(sys.stdin.readline().rstrip())
    n %= 30
    res = list(range(1, 7))
    for i in range(n):
      i %= 5
      res[i], res[i+1] = res[i+1], res[i]
    print(''.join(map(str, res)))

  def D(): pass 


class ABC005():
  def A():
    x, y = map(int, sys.stdin.readline().split())
    print(y//x)

  def B():
    n, *t = map(int, sys.stdin.read().split())
    print(min(t))

  def C():
    t = int(sys.stdin.readline().rstrip())
    n = int(sys.stdin.readline().rstrip())
    a = [int(x) for x in sys.stdin.readline().split()]
    m = int(sys.stdin.readline().rstrip())
    b = [int(x) for x in sys.stdin.readline().split()]
    i = 0
    for p in b:
      if i == n: print('no'); return 
      while p-a[i] > t:
        i += 1
        if i == n: print('no'); return 
      if a[i] > p: print('no'); return 
      i += 1
    print('yes')

  def D():
    n = int(sys.stdin.readline().rstrip())
    d = np.array([sys.stdin.readline().split() for _ in range(n)], np.int64)
    s = d.cumsum(axis=0).cumsum(axis=1)
    s = np.pad(s, 1)
    max_del = np.zeros((n+1, n+1), dtype=np.int64)
    for y in range(1, n+1):
      for x in range(1, n+1):
        max_del[y, x] = np.amax(s[y:n+1, x:n+1] - s[0:n-y+1, x:n+1] - s[y:n+1, 0:n-x+1] + s[0:n-y+1, 0:n-x+1])
    res = np.arange(n**2+1)[:, None]
    i = np.arange(1, n+1)
    res = max_del[i, np.minimum(res//i, n)].max(axis=1)
    q = int(sys.stdin.readline().rstrip())
    p = np.array(sys.stdin.read().split(), dtype=np.int64)
    print(*res[p], sep='\n')


class ABC006():
  def A():
    n = sys.stdin.readline().rstrip()
    if '3' in n: print('YES')
    elif int(n)%3 == 0: print('YES')
    else: print('NO')

  def B():
    mod = 10007
    t = [0, 0, 1]
    for _ in range(1001001):
      t.append(t[-1]+t[-2]+t[-3]); t[-1] %= mod
    n = int(sys.stdin.readline().rstrip())
    print(t[n-1])
    
  def C():
    n, m = map(int, sys.stdin.readline().split())
    cnt = [0, 0, 0]
    if m == 1: cnt = [-1, -1, -1]
    else:
      if m & 1: m -= 3; cnt[1] += 1; n -= 1
      cnt[2] = m//2 - n 
      cnt[0] = n - cnt[2]
    if cnt[0]<0 or cnt[1]<0 or cnt[2]<0: print(-1, -1, -1)
    else: print(*cnt, sep=' ')
      
  def D():
    n, *c = map(int, sys.stdin.read().split())
    lis = [inf]*n 
    for x in c: lis[bi_l(lis, x)] = x 
    print(n - bi_l(lis, inf))


class ABC007():
  def A():
    n = int(sys.stdin.readline().rstrip())
    print(n-1)

  def B():
    s = sys.stdin.readline().rstrip()
    if s == 'a': print(-1)
    else: print('a')

  def C():
    r, c = map(int, sys.stdin.readline().split())
    sy, sx = map(int, sys.stdin.readline().split())
    gy, gx = map(int, sys.stdin.readline().split())
    sy -= 1; sx -=1; gy -= 1; gx -= 1
    maze = [sys.stdin.readline().rstrip() for _ in range(r)]
    queue = deque([(sy, sx)])
    dist = np.full((r, c), np.inf); dist[sy, sx] = 0
    while queue:
      y, x = queue.popleft()
      for i, j in [(-1, 0), (1, 0), (0, -1), (0, 1)]:
        i += y; j += x
        if maze[i][j] == '#' or dist[i, j] != np.inf: continue 
        dist[i, j] = dist[y, x] + 1
        queue.append((i, j))
    print(int(dist[gy, gx]))
  
  def D(): pass 


class ABC008():
  def A():
    s, t = map(int, sys.stdin.readline().split())
    print(t-s+1)

  def B():
    n, *s = sys.stdin.read().split()
    res = defaultdict(int)
    for name in s: res[name] += 1
    print(sorted(res.items(), key=lambda x: x[1])[-1][0])

  def C():
    n, *a = map(int, sys.stdin.read().split())
    a = np.array(a)
    c = n - np.count_nonzero(a[:, None]%a, axis=1)
    print(np.sum((c+1)//2/c))
  
  def D(): pass

class ABC009():
  def A():
    n = int(sys.stdin.readline().rstrip())
    print((n+1)//2)

  def B():
    n, *a = map(int, sys.stdin.read().split())
    print(sorted(set(a))[-2])
    
  def C():
    n, k = map(int, sys.stdin.readline().split())
    s = list(sys.stdin.readline().rstrip())
    cost = [1]*n
    r = k
    for i in range(n-1):
      q = []
      for j in range(i+1, n):
        if s[j] < s[i] and cost[i]+cost[j] <= r:
          heappush(q, (s[j], cost[i]+cost[j], -j))
      if not q: continue
      _, c, j = heappop(q); j = -j 
      s[i], s[j] = s[j], s[i]
      r -= c 
      cost[i] = cost[j] = 0
    print(''.join(s))
      
  def D(): pass 


class ABC010():
  def A():
    print(sys.stdin.readline().rstrip()+'pp')
  def B():
    n, *a = map(int, sys.stdin.read().split())
    tot = 0
    for x in a:
      c = 0 
      while x%2==0 or x%3==2:
        x -= 1
        c += 1
      tot += c
    print(tot)

  def C():
    sx, sy, gx, gy, t, v, n, *xy = map(int, sys.stdin.read().split())
    x, y = np.array(xy).reshape(-1, 2).T
    def dist(x1, y1, x2, y2):
      return np.sqrt((x2-x1)**2 + (y2-y1)**2)
    ans = 'YES' if (dist(sx, sy, x, y)+dist(x, y, gx, gy) <= v*t).any() else 'NO'
    print(ans)

  def D(): pass 

class ABC011():
  def A():
    n = int(sys.stdin.readline().rstrip())
    print(n%12+1)

  def B():
    s = sys.stdin.readline().rstrip() 
    print(s[0].upper()+s[1:].lower())

  def C():
    n, *ng = map(int, sys.stdin.read().split())
    ng = set(ng)
    if n in ng: print('NO')
    else:
      r = 100
      while n > 0:
        if r == 0: print('NO'); return
        for i in range(3, 0, -1):
          if (n-i) in ng: continue 
          n -= i
          r -= 1
          break 
        else: print('NO'); return 
      print('YES')

  def D(): pass 


class ABC041():
  def A():
    s, i = sys.stdin.read().split()
    i = int(i)
    print(s[i-1])

  def B():
    a, b, c = map(int, sys.stdin.readline().split())
    ans = a * b % MOD * c % MOD 
    print(ans)

  def C():
    n, *a = map(int, sys.stdin.read().split())
    for i, h in sorted(enumerate(a), key=lambda x: -x[1]):
      print(i+1)

  def D():
    n, m, *xy = map(int, sys.stdin.read().split())
    *xy, = zip(*[iter(xy)]*2)
    edges = [0] * n 
    for x, y in xy:
      x -= 1; y -= 1
      edges[x] |= 1<<y
    comb = [None] * (1<<n); comb[0] = 1
    def count(edges, bit):
      if comb[bit] is not None: return comb[bit]
      comb[bit] = 0
      for i in range(n):
        if (bit>>i) & 1 and not edges[i]:
          nxt_bit = bit & ~(1<<i)
          nxt_edges = edges.copy() 
          for j in range(n):
            nxt_edges[j] &= ~(1<<i)
          cnt = count(nxt_edges, nxt_bit)
          comb[bit] += cnt
      return comb[bit]
    print(count(edges, (1<<n)-1))


class ABC042():
  def A():
    a = [int(x) for x in sys.stdin.readline().split()]
    c = Counter(a)
    print('YES' if c[5]==2 and c[7]==1 else 'NO')

  def B():
    n, l, *s = sys.stdin.read().split()
    print(''.join(sorted(s)))

  def C():
    n, k, *d = sys.stdin.read().split()
    l = len(n)
    ok = sorted(set(string.digits)-set(d))
    cand = [int(''.join(p)) for p in product(ok, repeat=l)] + [int(min(x for x in ok if x > '0')+min(ok)*l)]
    print(cand[bi_l(cand, int(n))])

  def D(): pass 
  h, w, a, b = map(int, sys.stdin.read().split())
  combinatorics = Combinatorics(mod=MOD)
  tot = combinatorics.mod_choose(h+w-2, h-1)
  for i in range(h-a, h):
    tot -= combinatorics.mod_choose(i+b-1, i) * combinatorics.mod_choose(h-i+w-b-2, h-1-i)
    tot %= MOD 
  print(tot)
  








class ABC170():
  def A():
    x = [int(x) for x in sys.stdin.readline().split()]
    for i in range(5):
      if x[i] != i+1:
        print(i+1)
        break

  def B():
    x, y = map(int, sys.stdin.readline().split())
    print('Yes' if 2*x <= y <= 4*x and y%2 == 0 else 'No')
      
  def C():
    x, n, *p = map(int, sys.stdin.read().split())
    a = list(set(range(102)) - set(p))
    a = [(abs(y-x), y) for y in a]
    print(sorted(a)[0][1])

  def D():
    n, *a = map(int, sys.stdin.read().split())
    cand = set(a)
    cnt = 0
    for x, c in sorted(Counter(a).items()):
      cnt += c == 1 and x in cand
      cand -= set(range(x*2, 10**6+1, x))
    print(cnt)

  def E():
    n, q = map(int, sys.stdin.readline().split())
    queue = [] 
    num_kindergarten = 2*10**5
    queue_kindergarten = [[] for _ in range(num_kindergarten)]
    highest_kindergarten = [None] * num_kindergarten
    where = [None] * n
    rate = [None] * n

    def entry(i, k):
      where[i] = k
      while queue_kindergarten[k]:
        r, j = heappop(queue_kindergarten[k])
        if where[j] != k or j == i: continue 
        if rate[i] >= -r:
          highest_kindergarten[k] = rate[i]
          heappush(queue, (rate[i], k, i))
        heappush(queue_kindergarten[k], (r, j))
        break
      else:
        highest_kindergarten[k] = rate[i]
        heappush(queue, (rate[i], k, i))
      heappush(queue_kindergarten[k], (-rate[i], i))

    def transfer(i, k):
      now = where[i]
      while queue_kindergarten[now]:
        r, j = heappop(queue_kindergarten[now])
        if where[j] != now or j == i: continue
        if highest_kindergarten[now] != -r:
          highest_kindergarten[now] = -r
          heappush(queue, (-r, now, j))
        heappush(queue_kindergarten[now], (r, j))
        break
      else:
        highest_kindergarten[now] = None
      entry(i, k)

    def inquire():
      while True:
        r, k, i = heappop(queue)
        if where[i] != k or r != highest_kindergarten[k]: continue 
        heappush(queue, (r, k, i))
        return r

    for i in range(n):
      a, b = map(int, sys.stdin.readline().split())
      rate[i] = a 
      entry(i, b-1)
    for _ in range(q):
      c, d = map(int, sys.stdin.readline().split())
      transfer(c-1, d-1)
      print(inquire())
    
  def F(): pass 


class ABC171():
  def A():
    c = sys.stdin.readline().rstrip()
    print('A' if c < 'a' else 'a')

  def B():
    n, k, *p = map(int, sys.stdin.read().split())
    print(sum(sorted(p)[:k]))

  def C():
    n = int(sys.stdin.readline().rstrip())
    n -= 1
    l = 1
    while True:
      if n < pow(26, l):
        break 
      n -= pow(26, l)
      l += 1
    res = ''.join([chr(ord('a')+d%26) for d in NumberTheory.base_convert(n, 26)][::-1])
    res = 'a'*(l-len(res)) + res
    print(res)

  def D():
    n = int(sys.stdin.readline().rstrip())
    a = [int(x) for x in sys.stdin.readline().split()]
    s = sum(a)
    cnt = Counter(a)
    q = int(sys.stdin.readline().rstrip())
    for _ in range(q):
      b, c = map(int, sys.stdin.readline().split())
      s += (c-b)*cnt[b]
      print(s)
      cnt[c] += cnt[b]; cnt[b] = 0

  def E():
    n, *a = map(int, sys.stdin.read().split())
    s = 0
    for x in a: s ^= x 
    b = map(lambda x: x^s, a)
    print(*b, sep=' ')

  def F(): pass 


class ABC172():
  def A(): pass
  def B(): pass
  def C(): pass 
  def D(): pass 
  def E(): pass 
  def F(): pass 


class ABC173():
  def A():
    n = int(sys.stdin.readline().rstrip())
    charge = (n+999)//1000 * 1000 - n
    print(charge)

  def B():
    n, *s = sys.stdin.read().split() 
    c = Counter(s)
    for v in 'AC, WA, TLE, RE'.split(', '):
      print(f'{v} x {c[v]}')

  def C():
    h, w, k = map(int, sys.stdin.readline().split())
    c = [sys.stdin.readline().rstrip() for _ in range(h)]
    tot = 0
    for i in range(1<<h):
      for j in range(1<<w):
        cnt = 0
        for y in range(h):
          for x in range(w):
            if i>>y & 1 or j>>x & 1:
              continue 
            cnt += c[y][x] ==  '#'
        tot += cnt == k
    print(tot)

  def D():
    n, *a = map(int, sys.stdin.read().split())
    a.sort(reverse=True)
    res = a[0] + sum(a[1:1+(n-2)//2])*2 + a[1+(n-2)//2]*(n & 1)
    print(res)

  def E():
    MOD = 10**9+7
    n, k, *a = map(int, sys.stdin.read().split())
    minus = [x for x in a if x < 0]
    plus = [x for x in a if x > 0]
    if len(plus) + len(minus)//2*2 >= k: # plus 
      *minus, = map(abs, minus)
      minus.sort(reverse=True)
      plus.sort(reverse=True)
      cand = []
      if len(minus)&1: minus = minus[:-1]
      for i in range(0, len(minus)-1, 2):
        cand.append(minus[i]*minus[i+1]%MOD)
      if k & 1:
        res = plus[0]
        plus = plus[1:]
      else:
        res = 1
      if len(plus)&1: plus = plus[:-1]
      for i in range(0, len(plus)-1, 2):
        cand.append(plus[i]*plus[i+1]%MOD)
      cand.sort(reverse=True)
      for x in cand[:k//2]:
        res *= x
        res %= MOD 
      print(res)
    elif 0 in a:
      print(0)
    else:
      cand = sorted(map(abs, a))
      res = 1
      for i in range(k):
        res *= cand[i]
        res %= MOD
      res = MOD - res
      print(res)
      pass
  
  def F(): pass 


if __name__ == '__main__':
  ABC042.D()

Submission status: Accepted

Solution code:
h,w,a,b=map(int,input().split())
fac=[0]*200001#iの階乗mod(1000000007)
inv=[0]*200001#iの逆元mod(1000000007)
fac[0]=1
ans=0
for i in range(1,200001):
    fac[i]=fac[i-1]*i%1000000007
inv[200000]=pow(fac[200000],1000000005,1000000007)
for i in range(199999,0,-1):
    inv[i]=(inv[i+1]*(i+1))%1000000007
    inv[0]=1
for i in range(h-a):
    if i==0:
        if h==1:
            x=1
        else:
            x=(fac[w-b+h-2
                   -i]*inv[w-1-b]*inv[h-1-i])%1000000007
    else:
        x=((fac[b-1+i]*inv[b-1]*inv[i])%1000000007)*((fac[w-b+h-2-i]*inv[w-b-1]*inv[h-1-i])%1000000007)
    ans=(ans+x)%1000000007
print(ans)

Submission status: Accepted

Solution code:
import sys
from functools import lru_cache
sys.setrecursionlimit(10**9)

def mul(x, y, mod):
  x %= mod
  y %= mod
  return x * y % mod

def div(x, y, mod):
  x %= mod
  y %= mod
  return x * mod_pow(y, mod - 2, mod) % mod

@lru_cache(maxsize=None)
def mod_pow(a, p, mod):
  if p == 0:
    return 1
  if p % 2 == 0:
    root = mod_pow(a, p / 2, mod)
    return root * root % mod
  else:
    return a * mod_pow(a, p - 1, mod) % mod

def dnm(n, mod, factorials):
  denominators = [0] * (n + 1)
  denominators[n] = div(1, factorials, mod)
  for i in reversed(range(n)):
    denominators[i] = mul(denominators[i+1], i+1, mod)
  return denominators

def nmr(n, mod):
  factorials = [0] * (n + 1)
  factorials[0] = 1
  for i in range(n):
    factorials[i + 1] = (i + 1) * factorials[i] % mod
  return factorials

def cmb(a, b, mod, factorials, factorial_divs):
    if len(factorials) == 0:
        raise
    af = factorials[a]
    bf = factorial_divs[b]
    abf = factorial_divs[a - b]
    return mul(mul(af, bf, mod), abf, mod)

MOD=10**9+7
H,W,A,B=map(int,input().split())
ans = 0

n = nmr(H+W-2, MOD)
d = dnm(H+W-2, MOD, n[H+W-2])

for i in range(H-A):
  x = cmb(B-1+i, B-1, MOD, n, d)
  y = cmb(H+W-3-(B-1+i), W-B-1, MOD, n, d)
  ans += x * y % MOD
  ans %= MOD

print(ans)

Submission status: Accepted

Solution code:
import sys
H,W,A,B = map(int,sys.stdin.readline().rstrip().split())
mod = 10**9+7

kaijou = [1]
for i in range(1,H+W-1):
    kaijou.append((kaijou[-1]*i) % mod)

a = 0
for i in range(B):
    b = kaijou[(H-A-1)+i]*pow(kaijou[i],mod-2,mod)*pow(kaijou[H-A-1],mod-2,mod)
    b %= mod
    b *= kaijou[(W+A-2)-i]*pow(kaijou[A-1],mod-2,mod)*pow(kaijou[W-1-i],mod-2,mod)
    b %= mod
    a += b
    a %= mod

c = kaijou[H+W-2]*pow(kaijou[H-1],mod-2,mod)*pow(kaijou[W-1],mod-2,mod)
c %= mod

print((c-a) % mod)

Submission status: Accepted

Solution code:
#!/usr/bin/python
# -*- coding: UTF-8 -*-

import sys


def get_ints():
    return map(int, sys.stdin.readline().strip().split())


def mul(x, y, mod):
    x %= mod
    y %= mod
    return x * y % mod


def div(x, y, mod):
    x %= mod
    y %= mod
    return x * mod_pow(y, mod - 2, mod) % mod


def mod_pow(a, p, mod):
    if p == 0:
        return 1
    if p % 2 == 0:
        root = mod_pow(a, p / 2, mod)
        return root * root % mod
    else:
        return a * mod_pow(a, p - 1, mod) % mod


def pre_factorial_divs(n, mod, factorials):
    factorial_divs = [0] * (n + 1)

    factorial_divs[n] = div(1, factorials, mod)
    for i in reversed(range(n)):
        factorial_divs[i] = mul(factorial_divs[i+1], i+1, mod)
    return factorial_divs


def pre_factorials(n, mod):
    factorials = [0] * (n + 1)
    factorials[0] = 1
    for i in range(n):
        factorials[i + 1] = (i + 1) * factorials[i] % mod
    return factorials


def fast_fast_fermat_comb(a, b, mod, factorials, factorial_divs):
    if len(factorials) == 0:
        raise
    af = factorials[a]
    bf = factorial_divs[b]
    abf = factorial_divs[a - b]
    return mul(mul(af, bf, mod), abf, mod)


def main():
    h, w, a, b = get_ints()

    ans = 0
    mod = 10 ** 9 + 7

    factorials = pre_factorials(h+w-2, mod)
    factorial_divs = pre_factorial_divs(h+w-2, mod, factorials[h+w-2])

    for i in range(h-a):
        x = fast_fast_fermat_comb(b-1+i, b-1, mod, factorials, factorial_divs)
        y = fast_fast_fermat_comb(h+w-3-(b-1+i), w-b-1, mod, factorials, factorial_divs)
        ans += x * y % mod
        ans %= mod

    print(ans)


if __name__ == "__main__":
    main()

Submission status: Accepted

Solution code:
h,w,a,b=map(int,input().split())

p=10**9+7
#p=127

def modp_factorial(n):
    s=1
    for x in range(1,h+1):
        s=(s*x) % p
    return s
def modp_prod(lst):
    s=1
    for x in lst:
        s=(s*x)%p
    return s
def inv(n):
    s=1
    q=p-2
    while q>0:
        if q&1:
            s=(s*n) % p
        n=(n*n) % p
        q>>=1
    return s

l=[1]
f=1
for x in range(1,h+w):
    f=(f*x) % p
    l.append(f)
invl=[inv(l[-1])]
for n in range(h+w-1,1,-1):
    invl.append((invl[-1]*n) % p)

invl.append(1)
invl.reverse()


s=0
for x in range(1,h-a+1):
    s=(s+modp_prod([l[x+b-2],invl[x-1],invl[b-1]\
        ,l[w-b+h-x-1],invl[h-x],invl[w-b-1]])) % p
print(s)

Submission status: Accepted

Solution code:
h,w,a,b = map(int,input().split())
mod = 10**9+7

MAX = 2*10**5
fact = [1]*(MAX+1)
for i in range(1, MAX+1):
    fact[i] = (fact[i-1]*i) % mod

inv = [1]*(MAX+1)
for i in range(2, MAX+1):
    inv[i] = inv[mod % i]*(mod-mod//i) % mod

fact_inv = [1]*(MAX+1)
for i in range(1, MAX+1):
    fact_inv[i] = fact_inv[i-1] * inv[i] % mod

def comb(n, k):
    if n < k:
        return 0
    return fact[n] * fact_inv[n-k] * fact_inv[k] % mod
  
ans = 0
for k in range(b+1,w+1):
  ans += comb(k+h-a-2,h-a-1)*comb(w-k+a-1,a-1)%mod
  ans %= mod
print(ans)

Submission status: Accepted

Solution code:
def cmb(n, r, p):
    if (r < 0) or (n < r):
        return 0
    r = min(r, n - r)
    return fact[n] * factinv[r] * factinv[n-r] % p

p = 10 ** 9 + 7
N = 3 * 10 ** 5  # N は必要分だけ用意する
fact = [1, 1]  # fact[n] = (n! mod p)
factinv = [1, 1]  # factinv[n] = ((n!)^(-1) mod p)
inv = [0, 1]  # factinv 計算用
 
for i in range(2, N + 1):
    fact.append((fact[-1] * i) % p)
    inv.append((-inv[p % i] * (p // i)) % p)
    factinv.append((factinv[-1] * inv[-1]) % p)
    
if __name__ == "__main__":
  H, W, A, B = map(int, input().split())
  #HW = [[0] * W for i in range(H)]
  p = 10 ** 9 + 7
  #print(HW)
  #visited = [[0] * W for i in range(H)]
  ans = 0
  for i in range(H - A):
    if i == 0:
      ans += cmb(B + i, i, p) * cmb(H + W - B - 2 - i, H - 1 - i, p)
      ans %= p
    else:
      ans += cmb(B + i - 1, i, p) * cmb(H + W - B - 2 - i, H - 1 - i, p)
      ans %= p
  print(ans)

Submission status: Accepted

Solution code:
import math
h, w, a, b = [int(i) for i in input().split()]

MOD = 10**9 + 7
N = h + w

fact = [1]*(N+1)
rfact = [1]*(N+1)
r = 1
for i in range(1, N+1):
  fact[i] = r = r * i % MOD
rfact[N] = r = pow(fact[N], MOD-2, MOD)
for i in range(N, 0, -1):
  rfact[i-1] = r = r * i % MOD

def comb(n, k):
  return fact[n] * rfact[k] * rfact[n-k] % MOD

tmp = []
for i in range(b, w):
    tmp.append(comb(i+h-a-1, i))

ans = 0
cnt = 0
for i in range(w-b-1, -1, -1):
    t = comb(i + a - 1, a - 1)
    ans += t * tmp[cnt]
    cnt += 1
print(ans%MOD)

Submission status: Accepted

Solution code:
h, w, a, b = map(int, input().split())

modulo = 10 ** 9 + 7

factorials = [1]
for i in range(1, h + w + 1):
    factorials.append((factorials[-1] * i) % modulo)


inverses = []
for f in factorials:
    inverses.append(pow(f, modulo - 2, modulo))

def cnk(n, k):
    return (factorials[n] * inverses[k] * inverses[n - k]) % modulo


line_before = [0] * w

for j in range(b, w):
    line_before[j] = cnk(h - a - 1 + j, h - a - 1)


s = 0
for j in range(b, w):
    s += cnk(a + w - j - 2, a - 1) * line_before[j]
    s %= modulo

print(s)

Submission status: Accepted

Solution code:
import sys
input = sys.stdin.readline
H,W,A,B= map(int,input().split())

MOD = 10**9 + 7
num = 200010
fact = [1] * (num+1)
ifact = [1] * (num+1)

for i in range(1,num+1):
    fact[i] = (fact[i-1] * i) % MOD
ifact[-1] = pow(fact[-1],MOD-2,MOD)
for i in range(1,num+1)[::-1]:
    ifact[i-1] = (ifact[i] * i)% MOD

def nCr(n,r):
    if r > n:
        return 0
    return (fact[n] * ifact[r] * ifact[n-r]) % MOD

ans = 0
for i in range(B+1,W+1):
    ans += nCr((i-1)+(H-A-1),i-1) * nCr((A-1)+(W-i),A-1)
    ans %= MOD
print(ans)

Submission status: Accepted

Solution code:
H, W, A, B = map(int, input().split())
C = H - A
D = W - B
p = 1000000007

def power(a, b): #a^b (mod p)を求める #二分累乗法を使う
    if b == 0:
        return 1
    elif b % 2 == 0:
        return power(a, b//2) ** 2 % p
    else:
        return power(a, b//2) ** 2 * a % p

f = [1] #f[i] = i! % p
for i in range(H+W):
    f.append(f[i] * (i+1) % p)

I_f = [0] * (H+W+1) #I_f[i] = (i!)^(-1) % p
I_f[H+W] = power(f[H+W], p-2)
for i in reversed(range(H+W)):
    I_f[i] = I_f[i+1] * (i+1) % p

def combi(a, b): #(a+b)!/a!b! (mod p)を求める
    return f[a+b] * I_f[a] * I_f[b] % p

x = 0
for i in range(C):
    x = (x + combi(i, B-1) * combi(D-1, H-i-1)) % p
print(x)

Submission status: Accepted

Solution code:
import sys

readline = sys.stdin.readline
MOD = 10 ** 9 + 7
INF = float('INF')
sys.setrecursionlimit(10 ** 5)


def comb_mod(n, r):
    res = 1
    r = min(n - r, r)

    for i in range(r):
        res *= (n - i)
        res %= MOD
        res *= pow((r - i), MOD - 2, MOD)

    return res


def main():
    h, w, a, b = map(int, readline().split())

    ans = 0
    comb1 = 1
    comb2 = comb_mod(w - b - 1 + h - 1, w - b - 1)

    for i in range(h - a):
        ans += comb1 * comb2
        ans %= MOD
        comb1 *= (b - 1 + i + 1)
        comb1 %= MOD
        comb1 *= pow(i + 1, MOD - 2, MOD)
        comb1 %= MOD
        comb2 *= pow(w - 1 - b + h - 1 - i, MOD - 2, MOD)
        comb2 %= MOD
        comb2 *= w - 1 - b + h- 1 - i - (w - b - 1)
        comb2 %= MOD

    print(ans)


if __name__ == '__main__':
    main()

Submission status: Accepted

Solution code:


def read_int():
    return int(input().strip())


def read_ints():
    return list(map(int, input().strip().split(' ')))


factorial = [1]
modulo = 10**9+7
inverse_factorial = [0]


def pow1(a, p):
    if p == 0:
        return 1
    half = pow1(a, p//2)
    total = (half*half)%modulo
    if p%2 == 0:
        return total
    return (total*a)%modulo


def modpow(a, p):
    ans = 1
    while p:
        if p&1 == 1:
            ans = (ans*a)%modulo
        a = (a*a)%modulo
        p >>= 1
    return ans


def nCr(n, r):
    if r == 0 or r == n:
        return 1
    return (((factorial[n]*inverse_factorial[n-r])%modulo)*inverse_factorial[r])%modulo


def solve():
    """
    2 3 1 1
    ...
    x..  A

    B

    C(2, 1)*C(3, 2)

    m 0 n 1 C(m, n) 1
    m 1 n 2 C(m, n) 2
    m 0 n 2 C(m, n) 4
    m 1 n 3 C(m, n) 12
    """
    H, W, A, B = read_ints()
    for i in range(1, H+W):
        f = (factorial[-1]*i)%modulo
        factorial.append(f)
        inv_f = modpow(f, modulo-2)
        inverse_factorial.append(inv_f)
    T = 0
    for d in range(B, W):
        # d = 0, 1
        y0 = d
        x0 = H-A-1
        y1 = W-d-1
        x1 = A-1
        T = (T+(nCr(x0+y0, x0)*nCr(x1+y1, x1))%modulo)%modulo
    return T


if __name__ == '__main__':
    print(solve())

Submission status: Accepted

Solution code:
H, W, A, B = map(int,input().split())
mod = 10**9+7
C = [1,1]
inv = [0,1]
Cinv = [1,1]
for i in range(2,H+W+1):
    C.append((C[-1]*i)%mod)
    inv.append((-inv[mod%i]*(mod//i)%mod))
    Cinv.append(Cinv[-1]*inv[-1]%mod)
ans = 0
for i in range(H-A):
    a = (C[i+B-1]*C[H-1-i+W-B-1])%mod
    a = (a*Cinv[B-1])%mod
    a = (a*Cinv[i])%mod
    a = (a*Cinv[W-B-1])%mod
    a = (a*Cinv[H-i-1])%mod
    ans += a
    ans %= mod
print(ans)