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AtCoder Beginner Contest 011 D: 大ジャンプ

解法

水平方向に飛ばなければならない回数を H 、垂直方向に飛ばなければならない回数を V とする。

垂直方向に飛ぶ回数を v とすると、垂直方向に戻る回数は v-V、 水平方向に飛ぶ回数 h=(N-v-v+V-H)/2+H、水平方向に戻る回数は h-H となる。この時ゴールに辿り着く確率は

 \frac{N!}{v!(v-V)!h!(h-H)!4^N}

である。これを各 v について計算すれば良い。

コード

import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.util.NoSuchElementException;

public class Main {
  private static class Task {
    void solve(FastScanner in, PrintWriter out) {
      int N = in.nextInt();
      int D = in.nextInt();
      int X = Math.abs(in.nextInt());
      int Y = Math.abs(in.nextInt());

      if (X % D != 0 || Y % D != 0) {
        out.println(0.0);
        return;
      }
      int V = X / D;
      int H = Y / D;
      double ans = 0.0;
      for (int v = V; v <= N; v++) {
        int antiV = v - V;
        int rest = N - v - antiV - H;
        if (rest < 0) continue;
        if (rest % 2 != 0) continue;
        int h = rest / 2 + H;
        int antiH = rest / 2;
        assert h + v + antiV + antiH == N;

        double tmp = 1.0;
        long n = 1;
        for (int i = 1; i <= h; i++, n++) {
          tmp *= n;
          tmp /= i;
          tmp /= 4.0;
        }

        for (int i = 1; i <= antiH; i++, n++) {
          tmp *= n;
          tmp /= i;
          tmp /= 4.0;
        }

        for (int i = 1; i <= v; i++, n++) {
          tmp *= n;
          tmp /= i;
          tmp /= 4.0;
        }

        for (int i = 1; i <= antiV; i++, n++) {
          tmp *= n;
          tmp /= i;
          tmp /= 4.0;
        }
        assert n == N;
        ans += tmp;
      }
      out.println(ans);
    }
  }

  /**
   * ここから下はテンプレートです。
   */
  public static void main(String[] args) {
    OutputStream outputStream = System.out;
    FastScanner in = new FastScanner();
    PrintWriter out = new PrintWriter(outputStream);
    Task solver = new Task();
    solver.solve(in, out);
    out.close();
  }
  private static class FastScanner {
    private final InputStream in = System.in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int bufferLength = 0;

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

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

    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();
    }

    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();
      }
    }

    double nextDouble() {
      return Double.parseDouble(next());
    }

    double[] nextDoubleArray(int n) {
      double[] array = new double[n];
      for (int i = 0; i < n; i++) {
        array[i] = nextDouble();
      }
      return array;
    }

    double[][] nextDoubleMap(int n, int m) {
      double[][] map = new double[n][];
      for (int i = 0; i < n; i++) {
        map[i] = nextDoubleArray(m);
      }
      return map;
    }

    public int nextInt() {
      return (int) nextLong();
    }

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

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

    public String[] nextStringArray(int n) {
      String[] array = new String[n];
      for (int i = 0; i < n; i++) array[i] = next();
      return array;
    }

    public char[][] nextCharMap(int n) {
      char[][] array = new char[n][];
      for (int i = 0; i < n; i++) array[i] = next().toCharArray();
      return array;
    }
  }
}