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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using static System.Console;
using static System.Math;

namespace AtTest.ABC_156
{
    class D
    {
        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(OpenStandardOutput()) { AutoFlush = false };
            SetOut(sw);

            Method(args);

            Out.Flush();
        }

        static void Method(string[] args)
        {
            long[] nab = ReadLongs();
            long n = nab[0];
            long a = nab[1];
            long b = nab[2];
            long mask = 1000000000 + 7;

            long[] pows = new long[40];
            pows[0] = 2;
            for (int i = 1; i < pows.Length; i++)
            {
                pows[i] = pows[i - 1] * pows[i - 1];
                pows[i] %= mask;
            }
            long nPow = 1;
            long tmp = 1;
            for (int i = 0; i < 40; i++)
            {
                if ((n / tmp) % 2 == 1)
                {
                    nPow *= pows[i];
                    nPow %= mask;
                }
                tmp *= 2;
            }
            long all = (nPow - 1 + mask) % mask;

            CaseCalculator calculator = new CaseCalculator(mask, Max(a, b));
            long nA = 1;
            long nB = 1;
            for (long i = n; i >= n - a + 1; i--)
            {
                nA *= i;
                nA %= mask;
            }
            for (long i = n; i >= n - b + 1; i--)
            {
                nB *= i;
                nB %= mask;
            }
            all = (all - (nA * calculator.Inverse(calculator.Permutation(a, a))) % mask + mask) % mask;
            all = (all - (nB * calculator.Inverse(calculator.Permutation(b, b))) % mask + mask) % mask;
            WriteLine(all);
        }

        class CaseCalculator
        {
            long mask;
            long[] allPermutations;
            long[] allInverses;

            public CaseCalculator(long mask, long permutationCnt)
            {
                this.mask = mask;
                allPermutations = AllPermutations(permutationCnt);
                allInverses = AllInverses(permutationCnt);
            }

            public long Combination(long n, long m)
            {
                if (n < m) return 0;

                if (n - m < m) m = n - m;

                return Multi(allPermutations[n], allInverses[n - m], allInverses[m]);
            }

            public long Permutation(long n, long m)
            {
                if (n < m) return 0;

                return Multi(allPermutations[n], allInverses[n - m]);
            }

            long[] AllPermutations(long n)
            {
                var perms = new long[n + 1];
                perms[0] = 1;
                for (int i = 1; i <= n; i++)
                {
                    perms[i] = Multi(perms[i - 1], i);
                }
                return perms;
            }

            long[] AllInverses(long n)
            {
                var inverses = new long[n + 1];
                inverses[1] = 1;
                var permInverses = new long[n + 1];
                permInverses[0] = 1;
                permInverses[1] = 1;
                for (int i = 2; i <= n; i++)
                {
                    inverses[i] = mask - inverses[mask % i] * (mask / i) % mask;
                    permInverses[i] = Multi(permInverses[i - 1], inverses[i]);
                }
                return permInverses;
            }

            public long Multi(params long[] vals)
            {
                if (vals.Length == 0)
                {
                    return 0;
                }

                long val = vals[0] % mask;
                for (int i = 1; i < vals.Length; i++)
                {
                    val *= (vals[i] % mask);
                    val %= mask;
                }
                return val;
            }

            public long Inverse(long val)
            {
                long a = mask;
                long b = val % mask;
                long x = 1;
                long x1 = 0;
                long y = 0;
                long y1 = 1;
                while (b > 0)
                {
                    long q = a / b;
                    long r = a % b;
                    long x2 = (mask + x - q * x1) % mask;
                    long y2 = (mask + y - q * y1) % mask;
                    a = b;
                    b = r;
                    x = x1;
                    x1 = x2;
                    y = y1;
                    y1 = y2;
                }
                return y;
            }

            long DeprecatedInverse(long val)
            {
                return Pow(val, mask - 2);
            }

            long Pow(long a, long exp)
            {
                if (exp == 0) return 1;
                else if (exp == 1) return a % mask;
                else
                {
                    long halfMod = Pow(a, exp / 2);
                    long nextMod = Multi(halfMod, halfMod);
                    if (exp % 2 == 0)
                    {
                        return nextMod;
                    }
                    else
                    {
                        return Multi(nextMod, a);
                    }
                }
            }
        }

        private static string Read() { return ReadLine(); }
        private static char[] ReadChars() { return Array.ConvertAll(Read().Split(), a => a[0]); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}