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 Rust (1.42.0) (judge verdict: Accepted)

#[allow(unused_imports)]
use itertools::Itertools;
#[allow(unused_imports)]
use num::*;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::collections::*;

#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct ModInt(u32);
impl ModInt {
    pub const MOD: u32 = 1_000_000_007;
    pub fn inv(self) -> Self {
        if self.0 == 0 {
            panic!();
        }
        self.pow(Self::MOD - 2)
    }
    pub fn one() -> Self {
        Self(1)
    }
    pub fn pow(self, e: u32) -> Self {
        if e == 0 {
            return Self::new(1);
        }
        let mut res = self.pow(e >> 1);
        res *= res;
        if e & 1 == 1 {
            res *= self;
        }
        res
    }
    pub fn zero() -> Self {
        Self(0)
    }
    fn new(n: i64) -> Self {
        let mut n = n % (Self::MOD as i64);
        if n.is_negative() {
            n += Self::MOD as i64;
        }
        Self(n as u32)
    }
}
impl From<i32> for ModInt {
    fn from(n: i32) -> Self {
        ModInt::from(n as i64)
    }
}
impl From<i64> for ModInt {
    fn from(n: i64) -> Self {
        Self::new(n)
    }
}
impl From<isize> for ModInt {
    fn from(n: isize) -> Self {
        ModInt::from(n as i64)
    }
}
impl From<u32> for ModInt {
    fn from(n: u32) -> Self {
        ModInt::from(n as u64)
    }
}
impl From<u64> for ModInt {
    fn from(n: u64) -> Self {
        Self::new(n as i64)
    }
}
impl From<usize> for ModInt {
    fn from(n: usize) -> Self {
        ModInt::from(n as u64)
    }
}
impl Into<i32> for ModInt {
    fn into(self) -> i32 {
        self.0 as i32
    }
}

impl Into<i64> for ModInt {
    fn into(self) -> i64 {
        self.0 as i64
    }
}

impl Into<isize> for ModInt {
    fn into(self) -> isize {
        self.0 as isize
    }
}

impl Into<u32> for ModInt {
    fn into(self) -> u32 {
        self.0
    }
}

impl Into<u64> for ModInt {
    fn into(self) -> u64 {
        self.0 as u64
    }
}

impl Into<usize> for ModInt {
    fn into(self) -> usize {
        self.0 as usize
    }
}

impl std::fmt::Display for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl std::ops::Add for ModInt {
    type Output = Self;

    fn add(self, rhs: Self) -> Self {
        Self::new((self.0 + rhs.0) as i64)
    }
}

impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl std::ops::Div for ModInt {
    type Output = Self;

    fn div(self, rhs: Self) -> Self {
        self * rhs.inv()
    }
}

impl std::ops::DivAssign for ModInt {
    fn div_assign(&mut self, rhs: Self) {
        *self = *self / rhs;
    }
}

impl std::ops::Mul for ModInt {
    type Output = Self;

    fn mul(self, rhs: Self) -> Self {
        Self::new((self.0 as i64) * (rhs.0 as i64))
    }
}

impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl std::ops::Sub for ModInt {
    type Output = Self;

    fn sub(self, rhs: Self) -> Self {
        Self::new((self.0 as i64) - (rhs.0 as i64))
    }
}

impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

fn nck(n: usize, k: usize) -> ModInt {
    let mut m = ModInt::one();
    let mut div_m = ModInt::one();
    for i in 0..k {
        m *= ModInt::from(n - i);
        div_m *= ModInt::from(i + 1);
    }
    m / div_m
}

fn solve() {
    input! {
        n: usize, a: usize, b: usize
    };

    let all = ModInt::from(2).pow(n as u32) - ModInt::one();
    let res = all - nck(n, a) - nck(n, b);
    println!("{}", res);
}

fn main() {
    std::thread::Builder::new()
        .name("big stack size".into())
        .stack_size(256 * 1024 * 1024)
        .spawn(|| {
            solve();
        })
        .unwrap()
        .join()
        .unwrap();
}