Buggy python submission (verdict: Runtime Error):
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 #出力の制限
N = 10**4
g1 = [1, 1] # 元テーブル
g2 = [1, 1] #逆元テーブル
inverse = [0, 1] #逆元テーブル計算用テーブル

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )

n,m,k = map(int,input().split())
nm = n*m%(10**9+7)
k = k%(10**9+7)
a = (cmb(nm-2,k-2,mod)%(10**9+7))*(nm*(m*(n**2-1)+n*(m**2-1)))//6
print(a%(10**9+7))

Fixed submission:
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 #出力の制限
N = 10**6
g1 = [1, 1] # 元テーブル
g2 = [1, 1] #逆元テーブル
inverse = [0, 1] #逆元テーブル計算用テーブル

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )

n,m,k = map(int,input().split())
nm = n*m%(10**9+7)
k = k%(10**9+7)
a = (cmb(nm-2,k-2,mod)%(10**9+7))*(nm*(m*(n**2-1)+n*(m**2-1)))//6
print(a%(10**9+7))

Buggy python submission (verdict: Time Limit Exceeded):
def reduce_mul(s,e):
     a = 1
     for i in range(s,e+1):
          a = a*i
     return a
     
def cmb(n,r):
    r = min(n-r,r)
    if r == 0: return 1
    over = reduce_mul(n-r+1,n)
    under = reduce_mul(1,r)
    return over // under

n,m,k = map(int,input().split())
nm = n*m%(10**9+7)
k = k%(10**9+7)
a = (cmb(nm-2,k-2)%(10**9+7))*(nm*(m*(n**2-1)+n*(m**2-1)))//6
print(a%(10**9+7))

Fixed submission:
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 #出力の制限
N = 10**6
g1 = [1, 1] # 元テーブル
g2 = [1, 1] #逆元テーブル
inverse = [0, 1] #逆元テーブル計算用テーブル

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )

n,m,k = map(int,input().split())
nm = n*m%(10**9+7)
k = k%(10**9+7)
a = (cmb(nm-2,k-2,mod)%(10**9+7))*(nm*(m*(n**2-1)+n*(m**2-1)))//6
print(a%(10**9+7))

Buggy python submission (verdict: Time Limit Exceeded):
def reduce_mul(s,e):
     a = 1
     for i in range(s,e+1):
          a = a*i
     return a
     
def cmb(n,r):
    r = min(n-r,r)
    if r == 0: return 1
    over = reduce_mul(n-r+1,n)
    under = reduce_mul(1,r)
    return over // under

n,m,k = map(int,input().split())
a = (cmb(n*m-2,k-2)%(10**9+7))*(n*m*(m*(n**2-1)+n*(m**2-1)))//6
print(a%(10**9+7))

Fixed submission:
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 #出力の制限
N = 10**6
g1 = [1, 1] # 元テーブル
g2 = [1, 1] #逆元テーブル
inverse = [0, 1] #逆元テーブル計算用テーブル

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )

n,m,k = map(int,input().split())
nm = n*m%(10**9+7)
k = k%(10**9+7)
a = (cmb(nm-2,k-2,mod)%(10**9+7))*(nm*(m*(n**2-1)+n*(m**2-1)))//6
print(a%(10**9+7))

Buggy python submission (verdict: Time Limit Exceeded):
from operator import mul
from functools import reduce

def cmb(n,r):
    r = min(n-r,r)
    if r == 0: return 1
    over = reduce(mul, range(n, n - r, -1))
    under = reduce(mul, range(1,r + 1))
    return over // under

def main():
     n,m,k = map(int,input().split())
     a = (cmb(n*m-2,k-2)%(10**9+7))*(n*m*(m*(n**2-1)+n*(m**2-1)))//6
     print(a%(10**9+7))
     return 0

if __name__=="__main__":
     main()

Fixed submission:
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 #出力の制限
N = 10**6
g1 = [1, 1] # 元テーブル
g2 = [1, 1] #逆元テーブル
inverse = [0, 1] #逆元テーブル計算用テーブル

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )

n,m,k = map(int,input().split())
nm = n*m%(10**9+7)
k = k%(10**9+7)
a = (cmb(nm-2,k-2,mod)%(10**9+7))*(nm*(m*(n**2-1)+n*(m**2-1)))//6
print(a%(10**9+7))

Buggy python submission (verdict: Time Limit Exceeded):
from sys import stdin
from operator import mul
from functools import reduce

def cmb(n,r):
    r = min(n-r,r)
    if r == 0: return 1
    over = reduce(mul, range(n, n - r, -1))
    under = reduce(mul, range(1,r + 1))
    return over // under

def main():
     n,m,k = [int(x) for x in stdin.readline().split()]
     a = cmb(n*m-2,k-2)*n*m*(m*(n**2-1)+n*(m**2-1))
     a = a//6
     a = a%(10**9+7)
     print(a)
     return

if __name__=="__main__":
     main()

Fixed submission:
def cmb(n, r, mod):
    if ( r<0 or r>n ):
        return 0
    r = min(r, n-r)
    return g1[n] * g2[r] * g2[n-r] % mod

mod = 10**9+7 #出力の制限
N = 10**6
g1 = [1, 1] # 元テーブル
g2 = [1, 1] #逆元テーブル
inverse = [0, 1] #逆元テーブル計算用テーブル

for i in range( 2, N + 1 ):
    g1.append( ( g1[-1] * i ) % mod )
    inverse.append( ( -inverse[mod % i] * (mod//i) ) % mod )
    g2.append( (g2[-1] * inverse[-1]) % mod )

n,m,k = map(int,input().split())
nm = n*m%(10**9+7)
k = k%(10**9+7)
a = (cmb(nm-2,k-2,mod)%(10**9+7))*(nm*(m*(n**2-1)+n*(m**2-1)))//6
print(a%(10**9+7))