Buggy python submission (verdict: Time Limit Exceeded):
from collections import defaultdict
def solve():
	Base = 2000
	MAX = Base*2
	p = 10**9+7
	F = [1]; Finv = [0]*(MAX*2+1)
	for i in xrange(MAX*2): F.append(F[i]*(i+1)%p)
	Finv[MAX] = pow(F[MAX],p-2,p)
	for i in xrange(MAX,0,-1): Finv[i-1] = Finv[i]*i%p
	N = int(raw_input())
	A_MAX = B_MAX = 0
	dp = [[0]*(MAX+2) for i in xrange(MAX+2)]
	goals = defaultdict(int)
	for i in xrange(N):
		A,B = map(int,raw_input().split())
		if A > A_MAX: A_MAX = A
		if B > B_MAX: B_MAX = B
		goals[(A,B)] += 1
		dp[Base-B][Base-A] += 1
	for j in xrange(Base-B_MAX,Base+B_MAX+1):
		for i in xrange(Base-A_MAX,Base+A_MAX+1):
			dp[j+1][i] = (dp[j+1][i] + dp[j][i]) % p
			dp[j][i+1] = (dp[j][i+1] + dp[j][i]) % p
	ans = 0
	for (A,B),v in goals.iteritems():
		ans = (ans+(dp[Base+B][Base+A]-F[A*2+B*2]*Finv[A*2]*Finv[B*2]%p)*v) % p
	print (ans + p)*pow(2,p-2,p) % p
solve()

Fixed submission:
from collections import defaultdict
def solve():
	Base = 2000
	MAX = Base*2
	p = 10**9+7
	F = [1]; Finv = [0]*(MAX*2+1)
	for i in xrange(MAX*2): F.append(F[i]*(i+1)%p)
	Finv[MAX] = pow(F[MAX],p-2,p)
	for i in xrange(MAX,0,-1): Finv[i-1] = Finv[i]*i%p
	N = int(raw_input())
	dup = A_MAX = B_MAX = 0
	dp = [[0]*(MAX+2) for i in xrange(MAX+2)]
	goals = defaultdict(int)
	for i in xrange(N):
		A,B = map(int,raw_input().split())
		dup = (dup + F[A*2+B*2]*Finv[A*2]*Finv[B*2])%p
		if A > A_MAX: A_MAX = A
		if B > B_MAX: B_MAX = B
		goals[(Base+A,Base+B)] += 1
		dp[Base-B][Base-A] += 1
	for j in xrange(Base-B_MAX,Base+B_MAX+1):
		for i in xrange(Base-A_MAX,Base+A_MAX+1):
			dp[j+1][i] = (dp[j+1][i] + dp[j][i]) % p
			dp[j][i+1] = (dp[j][i+1] + dp[j][i]) % p
	ans = 0
	for (i,j),v in goals.iteritems():
		ans = (ans + dp[j][i]*v) % p
	print ((ans - dup)*pow(2,p-2,p)+p) % p
solve()

Buggy python submission (verdict: Time Limit Exceeded):
from collections import defaultdict
def solve():
	Base = 2000
	MAX = Base*2
	p = 10**9+7
	F = [1]; Finv = [0]*(MAX*2+1)
	for i in xrange(MAX*2): F.append(F[i]*(i+1)%p)
	Finv[MAX] = pow(F[MAX],p-2,p)
	for i in xrange(MAX,0,-1): Finv[i-1] = Finv[i]*i%p
	N = int(raw_input())
	dup = A_MAX = B_MAX = 0
	dp = [[0]*(MAX+2) for i in xrange(MAX+2)]
	goals = defaultdict(int)
	for i in xrange(N):
		A,B = map(int,raw_input().split())
		dup = (dup + F[A*2+B*2]*Finv[A*2]*Finv[B*2])%p
		if A > A_MAX: A_MAX = A
		if B > B_MAX: B_MAX = B
		goals[(Base+A,Base+B)] += 1
		dp[Base-B][Base-A] += 1
	for j in xrange(Base-B_MAX,Base+B_MAX+1):
		for i in xrange(Base-A_MAX,Base+A_MAX+1):
			dp[j+1][i] = (dp[j+1][i] + dp[j][i]) % p
			dp[j][i+1] = (dp[j][i+1] + dp[j][i]) % p
	ans = 0
	for (i,j),v in goals.iteritems():
		ans = (ans + dp[j][i]*v) % p
	print (ans - dup + p)*pow(2,p-2,p) % p
solve()

Fixed submission:
from collections import defaultdict
def solve():
	Base = 2000
	MAX = Base*2
	p = 10**9+7
	F = [1]; Finv = [0]*(MAX*2+1)
	for i in xrange(MAX*2): F.append(F[i]*(i+1)%p)
	Finv[MAX] = pow(F[MAX],p-2,p)
	for i in xrange(MAX,0,-1): Finv[i-1] = Finv[i]*i%p
	N = int(raw_input())
	dup = A_MAX = B_MAX = 0
	dp = [[0]*(MAX+2) for i in xrange(MAX+2)]
	goals = defaultdict(int)
	for i in xrange(N):
		A,B = map(int,raw_input().split())
		dup = (dup + F[A*2+B*2]*Finv[A*2]*Finv[B*2])%p
		if A > A_MAX: A_MAX = A
		if B > B_MAX: B_MAX = B
		goals[(Base+A,Base+B)] += 1
		dp[Base-B][Base-A] += 1
	for j in xrange(Base-B_MAX,Base+B_MAX+1):
		for i in xrange(Base-A_MAX,Base+A_MAX+1):
			dp[j+1][i] = (dp[j+1][i] + dp[j][i]) % p
			dp[j][i+1] = (dp[j][i+1] + dp[j][i]) % p
	ans = 0
	for (i,j),v in goals.iteritems():
		ans = (ans + dp[j][i]*v) % p
	print ((ans - dup)*pow(2,p-2,p)+p) % p
solve()

Buggy python submission (verdict: Time Limit Exceeded):
from collections import defaultdict
def solve():
	Base = 2000
	MAX = Base*2
	p = 10**9+7
	F = [1]; Finv = [0]*(MAX*2+1)
	for i in xrange(MAX*2): F.append(F[i]*(i+1)%p)
	Finv[MAX*2] = pow(F[MAX*2],p-2,p)
	for i in xrange(MAX*2,0,-1): Finv[i-1] = Finv[i]*i%p
	N = int(raw_input())
	dup = A_MAX = B_MAX = 0
	dp = [[0]*(MAX+2) for i in xrange(MAX+2)]
	goals = defaultdict(int)
	for i in xrange(N):
		A,B = map(int,raw_input().split())
		dup = (dup + F[A*2+B*2]*Finv[A*2]*Finv[B*2]%p)%p
		if A > A_MAX: A_MAX = A
		if B > B_MAX: B_MAX = B
		goals[(Base+A,Base+B)] += 1
		dp[Base-B][Base-A] += 1
	for j in xrange(Base-B_MAX,Base+B_MAX+1):
		for i in xrange(Base-A_MAX,Base+A_MAX+1):
			dp[j+1][i] = (dp[j+1][i] + dp[j][i]) % p
			dp[j][i+1] = (dp[j][i+1] + dp[j][i]) % p
	ans = 0
	for (i,j),v in goals.iteritems():
		ans = (ans + dp[j][i]*v) % p
	print ((ans - dup)*pow(2,p-2,p) + p) % p
solve()

Fixed submission:
from collections import defaultdict
def solve():
	Base = 2000
	MAX = Base*2
	p = 10**9+7
	F = [1]; Finv = [0]*(MAX*2+1)
	for i in xrange(MAX*2): F.append(F[i]*(i+1)%p)
	Finv[MAX] = pow(F[MAX],p-2,p)
	for i in xrange(MAX,0,-1): Finv[i-1] = Finv[i]*i%p
	N = int(raw_input())
	dup = A_MAX = B_MAX = 0
	dp = [[0]*(MAX+2) for i in xrange(MAX+2)]
	goals = defaultdict(int)
	for i in xrange(N):
		A,B = map(int,raw_input().split())
		dup = (dup + F[A*2+B*2]*Finv[A*2]*Finv[B*2])%p
		if A > A_MAX: A_MAX = A
		if B > B_MAX: B_MAX = B
		goals[(Base+A,Base+B)] += 1
		dp[Base-B][Base-A] += 1
	for j in xrange(Base-B_MAX,Base+B_MAX+1):
		for i in xrange(Base-A_MAX,Base+A_MAX+1):
			dp[j+1][i] = (dp[j+1][i] + dp[j][i]) % p
			dp[j][i+1] = (dp[j][i+1] + dp[j][i]) % p
	ans = 0
	for (i,j),v in goals.iteritems():
		ans = (ans + dp[j][i]*v) % p
	print ((ans - dup)*pow(2,p-2,p)+p) % p
solve()

Buggy python submission (verdict: Time Limit Exceeded):
from collections import defaultdict
def solve():
	Base = 2000
	MAX = Base*2
	p = 10**9+7
	F = [1]; Finv = [0]*(MAX+1)
	for i in xrange(MAX*2): F.append(F[i]*(i+1)%p)
	Finv[MAX] = pow(F[MAX],p-2,p)
	for i in xrange(MAX,0,-1): Finv[i-1] = Finv[i]*i%p
	N = int(raw_input())
	dup = 0
	dp = [[0]*(MAX+2) for i in xrange(MAX+2)]
	goals = defaultdict(int)
	for i in xrange(N):
		A,B = map(int,raw_input().split())
		dup = (dup + F[A*2+B*2]*Finv[A*2]*Finv[B*2]%p)%p
		goals[(Base+A,Base+B)] += 1
		dp[Base-B][Base-A] += 1
	for j in xrange(MAX+1):
		for i in xrange(MAX+1):
			dp[j+1][i] += dp[j][i]
			dp[j][i+1] += dp[j][i]
	ans = 0
	for (i,j),v in goals.iteritems():
		ans = (ans + dp[j][i]*v) % p
	print ((ans - dup) / 2 + p) % p
solve()

Fixed submission:
from collections import defaultdict
def solve():
	Base = 2000
	MAX = Base*2
	p = 10**9+7
	F = [1]; Finv = [0]*(MAX*2+1)
	for i in xrange(MAX*2): F.append(F[i]*(i+1)%p)
	Finv[MAX] = pow(F[MAX],p-2,p)
	for i in xrange(MAX,0,-1): Finv[i-1] = Finv[i]*i%p
	N = int(raw_input())
	dup = A_MAX = B_MAX = 0
	dp = [[0]*(MAX+2) for i in xrange(MAX+2)]
	goals = defaultdict(int)
	for i in xrange(N):
		A,B = map(int,raw_input().split())
		dup = (dup + F[A*2+B*2]*Finv[A*2]*Finv[B*2])%p
		if A > A_MAX: A_MAX = A
		if B > B_MAX: B_MAX = B
		goals[(Base+A,Base+B)] += 1
		dp[Base-B][Base-A] += 1
	for j in xrange(Base-B_MAX,Base+B_MAX+1):
		for i in xrange(Base-A_MAX,Base+A_MAX+1):
			dp[j+1][i] = (dp[j+1][i] + dp[j][i]) % p
			dp[j][i+1] = (dp[j][i+1] + dp[j][i]) % p
	ans = 0
	for (i,j),v in goals.iteritems():
		ans = (ans + dp[j][i]*v) % p
	print ((ans - dup)*pow(2,p-2,p)+p) % p
solve()