Problem:
Score : 300 points

Problem Statement

How many integer sequences A_1,A_2,\ldots,A_N of length N satisfy all of the following conditions?

0 \leq A_i \leq 9

There exists some i such that A_i=0 holds.

There exists some i such that A_i=9 holds.

The answer can be very large, so output it modulo 10^9 + 7.

Constraints

1 \leq N \leq 10^6

N is an integer.

Input

Input is given from Standard Input in the following format:

N

Output

Print the answer modulo 10^9 + 7.

Sample Input 1

2

Sample Output 1

2

Two sequences \{0,9\} and \{9,0\} satisfy all conditions.

Sample Input 2

1

Sample Output 2

0

Sample Input 3

869121

Sample Output 3

2511445

Solution in Java (OpenJDK 11.0.6) (judge verdict: Accepted)



import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.util.Arrays;
import java.util.Random;
import java.util.StringTokenizer;

public final class Main {

    private static final int MOD = (int) (1e9 + 7);

    private static class Combinations {
        long[] factorial;
        long[] facInverse;
        long[] inverse;

        Combinations(int n) {
            final int MAX = n + 2;
            factorial = new long[MAX];
            facInverse = new long[MAX];
            inverse = new long[MAX];
            factorial[0] = factorial[1] = 1;
            facInverse[0] = facInverse[1] = 1;
            inverse[1] = 1;
            for (int i = 2; i < MAX; i++) {
                factorial[i] = factorial[i - 1] * i % MOD;
                final long inv = inverse[i] = MOD - inverse[MOD % i] * (MOD / i) % MOD;
                facInverse[i] = facInverse[i - 1] * inv % MOD;
            }
        }

        long ncr(int n, int r) {
            if (n < r) { return 0; }
            if (n < 0 || r < 0) { return 0; }
            return factorial[n] * (facInverse[r] * facInverse[n - r] % MOD) % MOD;
        }

        long modpow(long a, long n) {
            long res = 1;
            while (n > 0) {
                if (n % 2 == 1) {
                    res = res * a % MOD;
                }
                a = a * a % MOD;
                n /= 2;
            }
            return res;
        }
    }

    // https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle
    // Answer = (total sequences of length n with all elements <= 9) — (total sequences that do not contain 0 nor 9)
    // first term above is simply pow(10,n) for the second term I used inclusion exclusion:
    // A = Set of sequences that don't contain 0
    // B = Set of sequences that don't contain 9
    // n(A union B) = n(A) + n(B) - n(A intersection B)
    // n(A) = n(B) = pow(9,n)
    // n(A intersection B) = pow(8,n)
    public static void main(String[] args) {
        final FastScanner fs = new FastScanner();
        final int n = fs.nextInt();
        final Combinations combinations = new Combinations((int) (1e6 + 5));
        final long a = combinations.modpow(10, n);
        final long b = (2 * combinations.modpow(9, n)) % MOD;
        final long c = combinations.modpow(8, n);
        System.out.println(Math.floorMod(a - Math.floorMod(b - c, MOD), MOD));
    }

    static final class Util {
        public static void shuffleSort(int[] x) {
            shuffle(x);
            Arrays.sort(x);
        }

        public static void shuffleSort(long[] x) {
            shuffle(x);
            Arrays.sort(x);
        }

        public static void shuffle(int[] x) {
            final Random r = new Random();

            for (int i = 0; i <= x.length - 2; i++) {
                final int j = i + r.nextInt(x.length - i);
                swap(x, i, j);
            }
        }

        public static void shuffle(long[] x) {
            final Random r = new Random();

            for (int i = 0; i <= x.length - 2; i++) {
                final int j = i + r.nextInt(x.length - i);
                swap(x, i, j);
            }
        }

        public static void swap(int[] x, int i, int j) {
            final int t = x[i];
            x[i] = x[j];
            x[j] = t;
        }

        public static void swap(long[] x, int i, int j) {
            final long t = x[i];
            x[i] = x[j];
            x[j] = t;
        }

        private Util() {}
    }

    static class FastScanner {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        StringTokenizer st = new StringTokenizer("");

        private String next() {
            while (!st.hasMoreTokens()) {
                try {
                    st = new StringTokenizer(br.readLine());
                } catch (IOException e) {
                    //noinspection CallToPrintStackTrace
                    e.printStackTrace();
                }
            }
            return st.nextToken();
        }

        int nextInt() {
            return Integer.parseInt(next());
        }

        long nextLong() {
            return Long.parseLong(next());
        }

        int[] nextIntArray(int n) {
            final int[] a = new int[n];
            for (int i = 0; i < n; i++) { a[i] = nextInt(); }
            return a;
        }

        long[] nextLongArray(int n) {
            final long[] a = new long[n];
            for (int i = 0; i < n; i++) { a[i] = nextLong(); }
            return a;
        }
    }
}