Problem:
Score : 500 points

Problem Statement

Count the pairs of length-N sequences consisting of integers between 1 and M (inclusive), A_1, A_2, \cdots, A_{N} and B_1, B_2, \cdots, B_{N}, that satisfy all of the following conditions:

A_i \neq B_i, for every i such that 1\leq i\leq N.

A_i \neq A_j and B_i \neq B_j, for every (i, j) such that 1\leq i < j\leq N.

Since the count can be enormous, print it modulo (10^9+7).

Constraints

1\leq N \leq M \leq 5\times10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M

Output

Print the count modulo (10^9+7).

Sample Input 1

2 2

Sample Output 1

2

A_1=1,A_2=2,B_1=2,B_2=1 and A_1=2,A_2=1,B_1=1,B_2=2 satisfy the conditions.

Sample Input 2

2 3

Sample Output 2

18

Sample Input 3

141421 356237

Sample Output 3

881613484

Solution in Lua (LuaJIT 2.1.0) (judge verdict: Accepted)

local mod = 1000000007
local mfl = math.floor

local function bmul(x, y)
  local x1, y1 = mfl(x / 31623), mfl(y / 31623)
  local x0, y0 = x - x1 * 31623, y - y1 * 31623
  return (x1 * y1 * 14122 + (x1 * y0 + x0 * y1) * 31623 + x0 * y0) % mod
end

local function badd(x, y)
  return (x + y) % mod
end

local fact = {1}
local invs = {1}
local invfact = {1}

for i = 2, 500000 do
  fact[i] = bmul(fact[i - 1], i)
  invs[i] = bmul(mfl(mod / i), mod - invs[mod % i])
  invfact[i] = bmul(invfact[i - 1], invs[i])
end

local function getPerm(n, k)
  if n == 0 or k == 0 then return 1 end
  local ret = fact[n]
  if k < n then
    ret = bmul(ret, invfact[n - k])
  end
  return ret
end

local function getComb(n, k)
  if k == 0 or k == n then return 1 end
  return bmul(fact[n], bmul(invfact[k], invfact[n - k]))
end

local n, m = io.read("*n", "*n")
if m < n then
  print(0) os.exit()
end
local p_mn = getPerm(m, n)
local matched = 0
for i = 1, n do
  local comb = getComb(n, i)
  local remPerm = getPerm(m - i, n - i)
  local z = bmul(comb, remPerm)
  if i % 2 == 1 then
    matched = badd(matched, z)
  else
    matched = badd(matched, mod - z)
  end
end
local not_matched = badd(p_mn, mod - matched)
print(bmul(p_mn, not_matched))