Problem:
Score : 400 points

Problem Statement

We have an H-by-W matrix.
Let a_{ij} be the element at the i-th row from the top and j-th column from the left.
In this matrix, each a_{ij} is a lowercase English letter.

Snuke is creating another H-by-W matrix, A', by freely rearranging the elements in A.
Here, he wants to satisfy the following condition:

Every row and column in A' can be read as a palindrome.

Determine whether he can create a matrix satisfying the condition.

Note

A palindrome is a string that reads the same forward and backward.
For example, a, aa, abba and abcba are all palindromes, while ab, abab and abcda are not.

Constraints

1 ≤ H, W ≤ 100

a_{ij} is a lowercase English letter.

Input

Input is given from Standard Input in the following format:

H W
a_{11}a_{12}...a_{1W}
:
a_{H1}a_{H2}...a_{HW}

Output

If Snuke can create a matrix satisfying the condition, print Yes; otherwise, print No.

Sample Input 1

3 4
aabb
aabb
aacc

Sample Output 1

Yes

For example, the following matrix satisfies the condition.

abba
acca
abba

Sample Input 2

2 2
aa
bb

Sample Output 2

No

It is not possible to create a matrix satisfying the condition, no matter how we rearrange the elements in A.

Sample Input 3

5 1
t
w
e
e
t

Sample Output 3

Yes

For example, the following matrix satisfies the condition.

t
e
w
e
t

Sample Input 4

2 5
abxba
abyba

Sample Output 4

No

Sample Input 5

1 1
z

Sample Output 5

Yes

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)


import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.math.BigInteger;
import java.util.*;
import java.util.stream.Collectors;
import java.util.stream.IntStream;
import java.util.stream.LongStream;

public class Main {

    public Main() throws IOException {
    }

    public static void main(String[] args) throws IOException {
        new Main().solve();
    }

    //    private final InputStream in = Files.newInputStream(Paths.get("C:\\Users\\ryo.suzuki\\Downloads\\02.txt"));
    private final InputStream in = System.in;
    private final PrintWriter out = new PrintWriter(System.out);
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;

    private void solve() throws IOException {
        try {
//            solveA();
//            solveB();
//            solveC();
            solveD();
//            solveE();
//            solveF();
        } finally {
            if (in != null) {
                in.close();
            }
            if (out != null) {
                out.flush();
                out.close();
            }
        }

    }

    // mods
    long MOD = (long) Math.pow(10, 9) + 7;

    private void solveA() {
        int n = nextInt();
        out.println((int) Math.pow(n, 3));
    }

    private void solveB() {
        int n = nextInt();
        long[] a = LongStream.range(0, n).map(i -> nextLong()).toArray();
        Map<Long, Long> memo = new HashMap<Long, Long>();
        Arrays.stream(a).forEach(i -> {
                    memo.put(i + 1L, memo.getOrDefault(i + 1L, 0L) + 1L);
                    memo.put(i, memo.getOrDefault(i, 0L) + 1L);
                    memo.put(i - 1L, memo.getOrDefault(i - 1L, 0L) + 1L);
                }
        );
        Optional<Map.Entry<Long, Long>> res = memo.entrySet().stream().max((e1, e2) -> Long.compare(e1.getValue(), e2.getValue()));
        out.println(res.get().getValue());
    }

    private void solveC() {
        int n = nextInt();
        int m = nextInt();
        int[][] sc = IntStream.range(0, m).mapToObj(i -> new int[]{nextInt(), nextInt()}).toArray(int[][]::new);
        int[] res = new int[n];
        Arrays.fill(res, -1);
        for (int[] item : sc) {
            int idx = item[0] - 1;
            int val = item[1];
            if (res[idx] != -1 && res[idx] != val) {
                out.println(-1);
                return;
            }
            res[idx] = val;
        }
        if (n == 1 && res[0] <= 0) {
            out.println(0);
            return;
        }
        if (res[0] == -1)
            res[0] = 1;
        int outRes = new Integer(Arrays.stream(res).mapToObj(i -> i == -1 ? "0" : Integer.toString(i)).collect(Collectors.joining("")));

        int cnt = 0;
        int wk = outRes;
        while (wk > 0) {
            wk /= 10;
            cnt++;
        }
        if (cnt != n) {
            out.println(-1);
        } else {
            out.println(outRes);
        }
    }

    private void solveD() {
        int h = nextInt();
        int w = nextInt();
        char[][] a = IntStream.range(0, h).mapToObj(i -> next().toCharArray()).toArray(char[][]::new);
        /*
        a b c b a
        e f g f e
        a b c b a
        a b -> 4
        c f e -> 2(h/2+w/2以下)
        g -> 1

        a b c c b a
        e f g g f e
        e f g g f e
        a b c c b a
        a b c e f g -> 4

        a b c c b a
        e f g g f e
        a b c c b a
        a b c-> 4
        e f g -> 2(w/2以下)
         */
        int[] cnt = new int[26];
        Arrays.fill(cnt, -1);
        Arrays.stream(a).forEach(i -> {
            for (char item : i) {
                int idx = item - 'a';
                if (cnt[idx] == -1)
                    cnt[idx] = 0;
                cnt[idx]++;
            }
        });
        Map<Integer, Long> memo = Arrays.stream(cnt).filter(i -> i != -1).mapToObj(i -> i % 4).collect(Collectors.groupingBy(s -> s, Collectors.counting()));
        if (h % 2 == 0 && w % 2 == 0) {
            if (memo.containsKey(1) || memo.containsKey(2) || memo.containsKey(3)) {
                out.println("No");
            } else {
                out.println("Yes");
            }
        } else if (h % 2 == 1 && w % 2 == 1) {
            if (memo.getOrDefault(1, 1L) != 1L
                    || memo.getOrDefault(2, 0L) > (long) ((h - 1) / 2 + (w - 1) / 2)) {
                out.println("No");
            } else {
                out.println("Yes");
            }
        } else {
            int baseNum = (h % 2 == 0) ? h : w;
            if (memo.containsKey(1) || memo.containsKey(3) || memo.getOrDefault(2, 0L) > (long) (baseNum / 2)) {
                out.println("No");
            } else {
                out.println("Yes");
            }
        }

    }

    private void solveE() {

    }

    private void solveF() {

    }

    static final long CONST_INF = Long.MAX_VALUE / 4;

    public long lcmMod(long... ab) {

        Map<Long, Long> ps = new HashMap<Long, Long>();

        for (int abl = 0; abl < ab.length; abl++) {
            long target = ab[abl];
            long l = target;
            for (long p = 2; p * p <= l; p++) {
                long cnt = 0;
                while (target > 1 && target % p == 0) {
                    cnt++;
                    target /= p;
                }
                if (cnt == 0)
                    continue;
                ps.put(p, Long.max(cnt, ps.getOrDefault(p, 0L)));
            }
            if (target > 1)
                ps.put(target, Long.max(1, ps.getOrDefault(target, 0L)));
        }
        long res = 1L;
        for (Map.Entry<Long, Long> e : ps.entrySet()) {
            res *= powMod(e.getKey(), e.getValue());
            res %= MOD;
        }
        return res;
    }

    /**
     * 指定したx以下の素数をリストアップする
     */
    public long[] getPrimesUnderX(long x) {
        return LongStream.rangeClosed(2, x)
                .filter(i -> LongStream.rangeClosed(2, (long) Math.sqrt(i)).allMatch(j -> i % j != 0)).toArray();
    }

    public void partialUpdateAndWarshallFloyd(long[][] edges, int x, int y, long w) {
        int n = edges.length;
        if (edges[x][y] > w)
            edges[x][y] = edges[y][x] = w;
        int[] l = {x, y};
        for (int k : l) {
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < n; j++) {
                    if (edges[i][j] > edges[i][k] + edges[k][j]) {
                        edges[i][j] = edges[i][k] + edges[k][j];
                    }
                }
            }
        }
    }

    public long[][] createGraph(int n, int[][] edges) {
        long[][] graph = new long[n][n];
        for (int i = 0; i < n; i++) {
            Arrays.fill(graph[i], CONST_INF);
            graph[i][i] = 0;
        }
        int m = edges.length;
        for (int i = 0; i < m; i++) {
            int f = edges[i][0] - 1;
            int t = edges[i][1] - 1;
            long cost = edges[i][2];
            graph[f][t] = cost;
            graph[t][f] = cost;
        }
        return graph;
    }

    public long[][] warshallFloyd(int n, int[][] edges) {
        long[][] graph = createGraph(n, edges);
        for (int k = 0; k < n; k++) {
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < n; j++) {
                    graph[i][j] = Math.min(graph[i][j], graph[i][k] + graph[k][j]);
                }
            }
        }
        return graph;
    }

    private static class VectorCalculation {
        /**
         * 点Pと線(AB)の距離
         *
         * @param p
         * @param a
         * @param b
         * @return
         */
        private double distanceDotAndLine(DoublePair p, DoublePair a, DoublePair b) {
            DoublePair ab = new DoublePair(b.x - a.x, b.y - a.y);
            DoublePair ap = new DoublePair(p.x - a.x, p.y - a.y);

            // ベクトルAB、APの外積の絶対値が平行四辺形Dの面積になる
            double d = Math.abs(crossVector(ab, ap));

            double l = distanceVertex(a, b); // AB間の距離

            double h = d / l;
            return h;
        }

        /**
         * 2点間距離
         *
         * @param p1
         * @param p2
         * @return
         */
        private double distanceVertex(DoublePair p1, DoublePair p2) {
            double dx = p1.x - p2.x;
            double dy = p1.y - p2.y;
            return Math.sqrt(dx * dx + dy * dy);
        }

        /**
         * ベクトル外積
         *
         * @param vl
         * @param vr
         * @return
         */
        private double crossVector(DoublePair vl, DoublePair vr) {
            return vl.x * vr.y - vl.y * vr.x;
        }
    }

    private static class DoublePair implements Comparable<DoublePair> {
        double x;
        double y;

        public DoublePair(double k, double v) {
            this.x = k;
            this.y = v;
        }

        public int compareTo(DoublePair pair) {
            return Double.compare(this.y, pair.y);
        }
    }

    private static class SimplePair implements Comparable<SimplePair> {
        long k;
        long v;

        public SimplePair(int k, int v) {
            this.k = k;
            this.v = v;
        }

        public SimplePair(long k, long v) {
            this.k = k;
            this.v = v;
        }

        public int compareTo(SimplePair pair) {
            return Long.compare(this.v, pair.v);
        }
    }


    private static class SimpleGraphNode {
        private int n;
        private Map<Integer, Integer> childs;

        public SimpleGraphNode(int n) {
            this.n = n;
            childs = new HashMap<Integer, Integer>();
        }

        public void addChild(int child, int d) {
            this.childs.put(child, d);
        }
    }

    private static class SimpleGraph {

        private Map<Integer, SimpleGraphNode> graph;

        public SimpleGraph(int[][] list) {
            graph = new HashMap<Integer, SimpleGraphNode>();
            Arrays.stream(list).forEach(l -> {
                int a = l[0];
                int b = l[1];
                int d = l[2];
                graph.put(a, graph.getOrDefault(a, new SimpleGraphNode(a)));
                graph.get(a).addChild(b, d);
                graph.put(b, graph.getOrDefault(b, new SimpleGraphNode(b)));
                graph.get(b).addChild(a, d);
            });
        }

        public SimpleGraphNode getNode(int i) {
            return graph.getOrDefault(i, new SimpleGraphNode(-1));
        }

    }

    private static class GraphNode {
        private long n;
        private Map<Long, SimplePair> childs;

        public GraphNode(long n) {
            this.n = n;
            childs = new TreeMap<Long, SimplePair>();
        }

        /**
         * not update
         *
         * @param child
         * @param d
         */
        public void addChild(long child, long d) {
            this.childs.put(child, this.childs.getOrDefault(child, new SimplePair(child, d)));
        }

        public SimplePair getChild(long child) {
            return this.childs.getOrDefault(child, new SimplePair(child, CONST_INF));
        }
    }

    private static class Graph {

        private Map<Long, GraphNode> graph;
        int n;

        /**
         * @param n    頂点数
         * @param list
         */
        public Graph(int n, int[][] list) {
            this.n = n;
            graph = new HashMap<Long, GraphNode>();
            /*
            隣接行列作成
            0-indexed
             */
            Arrays.stream(list).forEach(l -> {
                long a = l[0] - 1;
                long b = l[1] - 1;
                long d = l[2];
                graph.put(a, graph.getOrDefault(a, new GraphNode(a)));
                graph.get(a).addChild(b, d);
                graph.put(b, graph.getOrDefault(b, new GraphNode(b)));
                graph.get(b).addChild(a, d);
            });
            /*
            たどり着けない場所はCONST_INFで埋める
            自分へはCOST=0でたどり着ける
             */
            for (long i = 0; i < n; i++) {
                graph.put(i, graph.getOrDefault(i, new GraphNode(i)));
                GraphNode node = graph.get(i);
                for (int j = 0; j < n; j++) {
                    if (i == j)
                        node.addChild(j, 0L);
                    else
                        node.addChild(j, CONST_INF);
                }
            }
        }

        /**
         * ワーシャルフロイド
         */
        public void warshallFloyd() {
            for (long k = 0; k < n; k++) {
                for (long i = 0; i < n; i++) {
                    for (long j = 0; j < n; j++) {
                        long min = Math.min(graph.get(i).getChild(j).v, graph.get(i).getChild(k).v + graph.get(k).getChild(j).v);
                        graph.get(i).getChild(j).v = min;
                    }
                }
            }
        }

        public void partialUpdateAndWarshallFloyd(long x, long y, long w) {
            if (graph.get(x).getChild(y).v > w)
                graph.get(x).getChild(y).v = graph.get(y).getChild(x).v = w;
            long[] l = {x, y};
            for (long k : l) {
                for (long i = 0; i < n; i++) {
                    for (long j = 0; j < n; j++) {
                        if (graph.get(i).getChild(j).v > graph.get(i).getChild(k).v + graph.get(k).getChild(j).v) {
                            graph.get(i).getChild(j).v = graph.get(i).getChild(k).v + graph.get(k).getChild(j).v;
                        }
                    }
                }
            }
        }

        /**
         * 2点間の距離
         *
         * @param from
         * @param to
         * @return
         */
        public long getDistance(long from, long to) {
            return graph.get(from).getChild(to).v;
        }

    }

    private static class SegTree {
        long e = 0;

        long calcFunc(long a, long b) {
            return a + b;
        }

        int n;
        long nodes[];

        public SegTree(long[] a) {
            init(a);
        }

        public SegTree(int len, long x) {
            long a[] = new long[len];
            Arrays.fill(a, x);
            init(a);
        }

        public SegTree(int len) {
            init(len);
        }

        void init(long[] a) {
            init(a.length);
            for (int i = 0; i < a.length; i++) {
                nodes[i + n - 1] = a[i];
            }
            for (int i = n - 2; i >= 0; i--) {
                nodes[i] = calcFunc(nodes[i * 2 + 1], nodes[i * 2 + 2]);
            }
        }

        void init(int len) {
            n = 1;
            while (n < len) {
                n *= 2;
            }
            nodes = new long[n * 2 - 1];
            Arrays.fill(nodes, e);
            for (int i = n - 2; i >= 0; i--) {
                nodes[i] = calcFunc(nodes[i * 2 + 1], nodes[i * 2 + 2]);
            }
        }

        void set(final int i, long val) {
            int wkI = i + n - 1;
            nodes[wkI] = val;
            while (wkI > 0) {
                wkI = (wkI - 1) / 2;
                nodes[wkI] = calcFunc(nodes[wkI * 2 + 1], nodes[wkI * 2 + 2]);
            }
        }

        long get(int i) {
            return find(i, i + 1);
        }

        /*
       要求区間 [a, b) 中の要素の最小値を答える
       k := 自分がいるノードのインデックス
       対象区間は [l, r) にあたる
        */
        long find(int a, int b) {
            return find(a, b, 0, 0, n);
        }

        long find(int a, int b, int k, int l, int r) {
            if (r < 0)
                r = n;

            if (b <= l || r <= a)
                return e;

            if (a <= l && r <= b)
                return nodes[k];

            long val1 = find(a, b, k * 2 + 1, l, (l + r) / 2);
            long val2 = find(a, b, k * 2 + 2, (l + r) / 2, r);
            //Math.min()
            //今回は区間内の個数なのでSUM
            return val1 + val2;
        }
    }

    private static class UnionFind {
        int[] pars;

        public UnionFind(int n) {
            pars = new int[n + 1];
            Arrays.fill(pars, -1);
        }

        public int getChilds(int n) {
            int wk = getRoot(n);
            return pars[wk];
        }

        public int getRoot(int n) {
            if (pars[n] < 0)
                return n;
            return pars[n] = getRoot(pars[n]);
        }

        public boolean isSame(int a, int b) {
            return getRoot(a) == getRoot(b);
        }

        private int getSize(int a) {
            return -pars[getRoot(a)];
        }

        public void unite(int a, int b) {
            int wkA = getRoot(a);
            int wkB = getRoot(b);
            if (wkA == wkB)
                return;

            if (getSize(wkA) < getSize(wkB)) {
                int tmp = wkA;
                wkA = wkB;
                wkB = tmp;
            }
            pars[wkA] = pars[wkA] + pars[wkB];
            pars[wkB] = wkA;
        }
    }

    public long calcSimpleCombination(long n, long m, long CONST_MOD) {
        long mole = 1;
        for (long i = 1; i <= n + m; i++) {
            mole *= i;
            mole %= CONST_MOD;
        }
        long deno = 1;
        for (long i = 1; i <= n; i++) {
            deno *= i;
            deno %= CONST_MOD;
        }
        for (int i = 1; i <= m; i++) {
            deno *= i;
            deno %= CONST_MOD;
        }
        deno = modInverse(deno, CONST_MOD);
        return (mole * deno) % CONST_MOD;
    }

    long abs(double x) {
        return (long) Math.abs(x);
    }

    long round(double x) {
        return Math.round(x);
    }

    long floor(double x) {
        return (long) Math.floor(x);
    }

    long ceil(double x) {
        return (long) Math.ceil(x);
    }

    double sqrt(double x) {
        return Math.sqrt(x);
    }

    double pow(double x, double y) {
        return Math.pow(x, y);
    }

    long pow(long x, long y) {
        return (long) Math.pow(x, y);
    }

    int gcd(int a, int b) {
        return b == 0 ? a : gcd(b, a % b);
    }

    long gcd(long a, long b) {
        return b == 0 ? a : gcd(b, a % b);
    }

    long lcm(long a, long b) {
        return a * b / gcd(a, b);
    }

    int upperToInt(char a) {
        return a - 'A';
    }

    int lowerToInt(char a) {
        return a - 'a';
    }

    int numToInt(char a) {
        return a - '0';
    }

    int charToInt(char a) {
        return a >= 'a' ? lowerToInt(a) : a >= 'A' ? upperToInt(a) : numToInt(a);
    }

    char intToUpper(int a) {
        return (char) (a + 'A');
    }

    char intToLower(int a) {
        return (char) (a + 'a');
    }

    char intToNum(int a) {
        return (char) (a + '0');
    }

    void reverse(String array[]) {
        String reversed[] = new String[array.length];
        for (int i = 0; i < array.length; i++) {
            reversed[array.length - i - 1] = array[i];
        }
        for (int i = 0; i < array.length; i++) {
            array[i] = reversed[i];
        }
    }

    void reverse(int array[]) {
        int reversed[] = new int[array.length];
        for (int i = 0; i < array.length; i++) {
            reversed[array.length - i - 1] = array[i];
        }
        for (int i = 0; i < array.length; i++) {
            array[i] = reversed[i];
        }
    }

    void reverse(long array[]) {
        long reversed[] = new long[array.length];
        for (int i = 0; i < array.length; i++) {
            reversed[array.length - i - 1] = array[i];
        }
        for (int i = 0; i < array.length; i++) {
            array[i] = reversed[i];
        }
    }

    void reverse(double array[]) {
        double reversed[] = new double[array.length];
        for (int i = 0; i < array.length; i++) {
            reversed[array.length - i - 1] = array[i];
        }
        for (int i = 0; i < array.length; i++) {
            array[i] = reversed[i];
        }
    }

    void reverse(boolean array[]) {
        boolean reversed[] = new boolean[array.length];
        for (int i = 0; i < array.length; i++) {
            reversed[array.length - i - 1] = array[i];
        }
        for (int i = 0; i < array.length; i++) {
            array[i] = reversed[i];
        }
    }

    void fill(int array[], int x) {
        Arrays.fill(array, x);
    }

    void fill(long array[], long x) {
        Arrays.fill(array, x);
    }

    void fill(double array[], double x) {
        Arrays.fill(array, x);
    }

    void fill(boolean array[], boolean x) {
        Arrays.fill(array, x);
    }

    void fill(int array[][], int x) {
        for (int a[] : array) {
            fill(a, x);
        }
    }

    void fill(long array[][], long x) {
        for (long a[] : array) {
            fill(a, x);
        }
    }

    void fill(double array[][], double x) {
        for (double a[] : array) {
            fill(a, x);
        }
    }

    void fill(boolean array[][], boolean x) {
        for (boolean a[] : array) {
            fill(a, x);
        }
    }

    void fill(int array[][][], int x) {
        for (int a[][] : array) {
            fill(a, x);
        }
    }

    void fill(long array[][][], long x) {
        for (long a[][] : array) {
            fill(a, x);
        }
    }

    void fill(double array[][][], double x) {
        for (double a[][] : array) {
            fill(a, x);
        }
    }

    void fill(boolean array[][][], boolean x) {
        for (boolean a[][] : array) {
            fill(a, x);
        }
    }

    long INF = (long) 1e18 + 7;

    boolean isINF(long a) {
        return abs(a) > INF / 1000;
    }

    boolean isPlusINF(long a) {
        return a > 0 && isINF(a);
    }

    boolean isMinusINF(long a) {
        return isPlusINF(-a);
    }

    int I_INF = (int) 1e9 + 7;

    boolean isINF(int a) {
        return abs(a) > I_INF / 1000;
    }

    boolean isPlusINF(int a) {
        return a > 0 && isINF(a);
    }

    boolean isMinusINF(int a) {
        return isPlusINF(-a);
    }

    public long mod(long i) {
        return i % MOD + ((i % MOD) < 0 ? MOD : 0);
    }

    long powMod(long x, long y) {
        if (y == 0) {
            return 1;
        } else {
            long tmp = powMod(x, y / 2);
            return mod(mod(tmp * tmp) * (y % 2 == 0 ? 1 : x));
        }
    }

    long[] powsMod(long x, int max) {
        long pow[] = new long[max + 1];
        pow[0] = 1;
        for (int i = 0; i < max; i++) {
            pow[i + 1] = mod(pow[i] * x);
        }
        return pow;
    }

    long inv(long x) {
        return powMod(x, MOD - 2);
    }

    int MAX_FACT = 5_000_100;
    long fact[];
    long invFact[];

    /**
     * Combination簡易版
     * 5 C 2
     * 異なる n個のものから r個を選ぶ組み合わせの総数 nCr を求めます。
     * 5!(5*4*3*2*1)
     * /
     * 2!(2*1) *  (5-2)!(3*2*1)
     *
     * @param n
     * @param r
     * @return
     */
    private long getComb(int n, int r) {
        long tmp = 1;
        for (int i = 1; i <= r; i++) {
            tmp *= n - i + 1;
            tmp = mod(tmp);
            tmp *= inv(i);
            tmp = mod(tmp);
        }
        return tmp;
    }

    /**
     * 階乗計算の事前累積和
     * [1, 1, 2, 3, 4, 5, … FACTORIAL_NUM]
     * mod済
     */
    void prepareFact() {
        fact = new long[MAX_FACT];
        Arrays.fill(fact, 0);
        invFact = new long[MAX_FACT];
        Arrays.fill(invFact, 0);
        fact[0] = 1;
        int maxIndex = (int) min(MAX_FACT, (int) MOD);
        for (int i = 1; i < maxIndex; i++) {
            fact[i] = mod(fact[i - 1] * i);
        }
        invFact[maxIndex - 1] = inv(fact[maxIndex - 1]);
        for (int i = maxIndex - 1; i > 0; i--) {
            invFact[i - 1] = mod(invFact[i] * i);
        }
    }

    /**
     * 順列
     * nPk -> n! / (n-k)!
     *
     * @param n
     * @param r
     * @return
     */
    long permutation(int n, int r) {
        if (n < 0 || r < 0 || n < r) {
            return 0;
        }
        return mod(fact[n] * invFact[n - r]);
    }

    /**
     * 組み合わせ
     * nCk -> n! / k!・(n-k)!
     *
     * @param n
     * @param r
     * @return
     */
    long combination(int n, int r) {
        if (n < 0 || r < 0 || n < r) {
            return 0;
        }
        return mod(permutation(n, r) * invFact[r]);
    }

    /**
     * 重複組合せ nHr （同次積）
     * nHr = (n+r-1)Cr
     * 異なるn個のものから重複を許してr個取る組合せの総数
     * 例：
     * リンゴ，ミカン，牛肉の3種類の果物があります．これらの中から6個の食材を買って帰ります．
     * このとき，何通りの買い方がありますか？ただし，含まれない食材があってもよいものとします
     *
     * @param n
     * @param r
     * @return
     */
    long homogeneousProduct(int n, int r) {
        return combination((n - 1) + r, r);
    }

    /**
     * 多項係数
     * 文字aをp個,bをq個,cをr個, dをs個 あわせてn個を1列に並べるときの並べ方
     * n! / p!・q!・r!・s!
     *
     * @param n
     * @param strNum
     * @param mod
     * @return
     */

    /**
     * フェルマーの小定理を用いて逆元を求める。
     * ある数xのmod p（pは素数）の逆数x'はx' = x^(p-2)
     * 繰り返し二乗法を用いて計算する。
     * http://satanic0258.hatenablog.com/entry/2016/04/29/004730
     * {@link BigInteger#modInverse(BigInteger)}とどちらが速いか？
     *
     * @param x
     * @return
     */
    private long modInverse(long x, long mod) {
        long res = 1L;
        long k = mod - 2L;
        long y = x;
        while (k != 0) {
            if (k % 2 != 0) {
                res = (res * y) % mod;
            }
            y = (y * y) % mod;
            k /= 2;
        }
        return res;
    }

    private boolean hasNextByte() {
        if (ptr < buflen) {
            return true;
        } else {
            ptr = 0;
            try {
                buflen = in.read(buffer);
            } catch (IOException e) {
                e.printStackTrace();
            }
            if (buflen <= 0) {
                return false;
            }
        }
        return true;
    }

    private int readByte() {
        if (hasNextByte())
            return buffer[ptr++];
        else
            return -1;
    }

    private static boolean isPrintableChar(int c) {
        return 33 <= c && c <= 126;
    }

    private void skipUnprintable() {
        while (hasNextByte() && !isPrintableChar(buffer[ptr]))
            ptr++;
    }

    public boolean hasNext() {
        skipUnprintable();
        return hasNextByte();
    }

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

    public String next() {
        if (!hasNext())
            throw new NoSuchElementException();
        StringBuilder sb = new StringBuilder();
        int b = readByte();
        while (isPrintableChar(b)) {
            sb.appendCodePoint(b);
            b = readByte();
        }
        return sb.toString();
    }

    public long nextLong() {
        if (!hasNext())
            throw new NoSuchElementException();
        long n = 0;
        boolean minus = false;
        int b = readByte();
        if (b == '-') {
            minus = true;
            b = readByte();
        }
        if (b < '0' || '9' < b) {
            throw new NumberFormatException();
        }
        while (true) {
            if ('0' <= b && b <= '9') {
                n *= 10;
                n += b - '0';
            } else if (b == -1 || !isPrintableChar(b)) {
                return minus ? -n : n;
            } else {
                throw new NumberFormatException();
            }
            b = readByte();
        }
    }

    public long min(long... v) {
        long min = Long.MAX_VALUE;
        for (long i : v) min = Math.min(min, i);
        return min;
    }

    public long max(long... v) {
        long max = Long.MIN_VALUE;
        for (long i : v) max = Math.max(max, i);
        return max;
    }
}