Problem:
Score : 400 points

Problem Statement

Akari has n kinds of flowers, one of each kind.

She is going to choose one or more of these flowers to make a bouquet.

However, she hates two numbers a and b, so the number of flowers in the bouquet cannot be a or b.

How many different bouquets are there that Akari can make?

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

Here, two bouquets are considered different when there is a flower that is used in one of the bouquets but not in the other bouquet.

Constraints

All values in input are integers.

2 \leq n \leq 10^9

1 \leq a < b \leq \textrm{min}(n, 2 \times 10^5)

Input

Input is given from Standard Input in the following format:

n a b

Output

Print the number of bouquets that Akari can make, modulo (10^9 + 7). (If there are no such bouquets, print 0.)

Sample Input 1

4 1 3

Sample Output 1

7

In this case, Akari can choose 2 or 4 flowers to make the bouquet.

There are 6 ways to choose 2 out of the 4 flowers, and 1 way to choose 4, so there are a total of 7 different bouquets that Akari can make.

Sample Input 2

1000000000 141421 173205

Sample Output 2

34076506

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

Solution in Haskell (GHC 7.10.3) (judge verdict: Wrong Answer)


import qualified Data.Vector.Unboxed as U
import qualified Data.Vector as V
import qualified Data.ByteString.Char8 as C
import Debug.Trace
import Data.List
import Control.Monad
import Data.Ord
import qualified Data.Sequence as Seq
import qualified Data.IntSet as IS
import qualified Data.Set as S

r :: Int -> IO (U.Vector Int)
r n = U.unfoldrN n (C.readInt . C.dropWhile (==' ')) <$> C.getLine

r' :: IO [Int]
r'  =   unfoldr    (C.readInt . C.dropWhile (==' ')) <$> C.getLine

rp :: IO (Int,Int)
rp  = toTup . unfoldr    (C.readInt . C.dropWhile (==' ')) <$> C.getLine

toTup [a,b] = (a,b)


mdl :: Num a => a
mdl = 10^9 + 7

main :: IO ()
main = do
    [n,a,b] <- r' :: IO [Int]
    let al = powMod 2 n - 1
    let na = choose n a `mod` mdl
    let nb = choose n b `mod` mdl
    -- print al
    -- print na
    -- print nb
    print $ positivize $ al - na - nb


powMod = powModAux 1
powModAux :: Int -> Int -> Int -> Int
powModAux ac cur 0 = ac
powModAux ac cur n
    | even n = powModAux ac (cur^2 `mod` mdl) (div n 2)
    | odd n  = powModAux (ac*cur `mod` mdl) (cur^2 `mod` mdl) (div n 2)



positivize x = if x < 0 then x + mdl else x


choose :: Int -> Int -> Int
choose n i = (prod' $ U.generate i (+ (n-i+1)) ) * (prodInv' $ U.generate i succ)

prod' :: U.Vector Int -> Int
prod' = U.foldl' (\ac x -> ac * x `mod` mdl) 1

prodInv' :: U.Vector Int -> Int
prodInv' = U.foldl' (\ac x -> ac * (modInv x) `mod` mdl) 1



modInv :: Int -> Int
modInv = (`powMod` (mdl-2) )
    -- let (x,_,1) = eEA a mdl
    -- in  positivize x

eEA :: Int -> Int -> (Int, Int, Int)
eEA a 0 = (1, 0, a)
eEA a b =
    let (q, r) = a `quotRem` b
        (s, t, g) = eEA b r
    in  (t, s-q*t, g)