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Number of ways to reach (X, Y) in a matrix starting from the origin

  • Difficulty Level : Hard
  • Last Updated : 15 Jun, 2021

Given two integers X and Y. The task is to find the number of ways to reach (X, Y) in a matrix starting from the origin when the possible moves are from (i, j) to either (i + 1, j + 2) or (i + 2, j + 1). Rows are numbered from top to bottom and columns are numbered from left to right. The answer could be large, so print the answer modulo 109 + 7

Examples:  

Input: X = 3, Y = 3 
Output:
The only possible ways are (0, 0) -> (1, 2) -> (3, 3) 
and (0, 0) -> (2, 1) -> (3, 3)

Input: X = 2, Y = 3 
Output:

Approach: The value of x coordinate + y coordinate increases by 3 with one movement. So when X + Y is not a multiple of 3, the answer is 0. When the number of movements of (+1, +2) is n and the number of movements of (+2, +1) is m then n + 2m = X, 2n + m = Y. The answer is 0 when n < 0 or m < 0. If not, the answer is n + m C n because it is only necessary to decide which n + 1 of the total n + m moves (+ 1, + 2). This value can be calculated by O(n + m + log mod) by calculating the factorial and its inverse. It can also be calculated with O(min {n, m}).



Below is the implementation of the above approach:

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
#define N 1000005
#define mod (int)(1e9 + 7)
 
// To store the factorial and factorial
// mod inverse of the numbers
int factorial[N], modinverse[N];
 
// Function to find (a ^ m1) % mod
int power(int a, int m1)
{
    if (m1 == 0)
        return 1;
    else if (m1 == 1)
        return a;
    else if (m1 == 2)
        return (1LL * a * a) % mod;
    else if (m1 & 1)
        return (1LL * a * power(power(a, m1 / 2), 2)) % mod;
    else
        return power(power(a, m1 / 2), 2) % mod;
}
 
// Function to find the factorial
// of all the numbers
void factorialfun()
{
    factorial[0] = 1;
    for (int i = 1; i < N; i++)
        factorial[i] = (1LL * factorial[i - 1] * i) % mod;
}
 
// Function to find the factorial
// modinverse of all the numbers
void modinversefun()
{
    modinverse[N - 1] = power(factorial[N - 1], mod - 2) % mod;
 
    for (int i = N - 2; i >= 0; i--)
        modinverse[i] = (1LL * modinverse[i + 1] * (i + 1)) % mod;
}
 
// Function to return nCr
int binomial(int n, int r)
{
    if (r > n)
        return 0;
 
    int a = (1LL * factorial[n]
             * modinverse[n - r])
            % mod;
 
    a = (1LL * a * modinverse[r]) % mod;
    return a;
}
 
// Function to return the number of ways
// to reach (X, Y) in a matrix with the
// given moves starting from the origin
int ways(int x, int y)
{
    factorialfun();
    modinversefun();
 
    if ((2 * x - y) % 3 == 0
        && (2 * y - x) % 3 == 0) {
        int m = (2 * x - y) / 3;
        int n = (2 * y - x) / 3;
        return binomial(n + m, n);
    }
 
    return 0;
}
 
// Driver code
int main()
{
    int x = 3, y = 3;
 
    cout << ways(x, y);
 
    return 0;
}

Java




// Java implementation of the approach
import java.util.*;
 
class GFG{
 
// To store the factorial and factorial
// mod inverse of the numbers
static long []factorial = new long [1000005];
static long  []modinverse = new long[1000005];
static long mod = 1000000007;
static int N = 1000005;
 
// Function to find (a ^ m1) % mod
static long power(long a, long m1)
{
    if (m1 == 0)
        return 1;
    else if (m1 == 1)
        return a;
    else if (m1 == 2)
        return (a * a) % mod;
    else if ((m1 & 1) != 0)
        return (a * power(power(a, m1 / 2), 2)) % mod;
    else
        return power(power(a, m1 / 2), 2) % mod;
}
 
// Function to find the factorial
// of all the numbers
static void factorialfun()
{
    factorial[0] = 1;
     
    for(int i = 1; i < N; i++)
        factorial[i] = (factorial[i - 1] * i) % mod;
}
 
// Function to find the factorial
// modinverse of all the numbers
static void modinversefun()
{
    modinverse[N - 1] = power(factorial[N - 1],
                                      mod - 2) % mod;
 
    for(int i = N - 2; i >= 0; i--)
        modinverse[i] = (modinverse[i + 1] *
                                   (i + 1)) % mod;
}
 
// Function to return nCr
static long binomial(int n, int r)
{
    if (r > n)
        return 0;
 
    long a = (factorial[n] *
             modinverse[n - r]) % mod;
 
    a = (a * modinverse[r]) % mod;
    return a;
}
 
// Function to return the number of ways
// to reach (X, Y) in a matrix with the
// given moves starting from the origin
static long ways(long x, long y)
{
    factorialfun();
    modinversefun();
 
    if ((2 * x - y) % 3 == 0 &&
        (2 * y - x) % 3 == 0)
    {
        long m = (2 * x - y) / 3;
        long n = (2 * y - x) / 3;
         
        // System.out.println(n+m+" "+n);
        return binomial((int)(n + m), (int)n);
    }
    return 0;
}
 
// Driver code
public static void main(String[] args)
{
    long x = 3, y = 3;
 
    System.out.println(ways(x, y));
}
}
 
// This code is contributed by Stream_Cipher

Python3




# Python3 implementation of the approach
N = 1000005
mod = (int)(1e9 + 7)
 
# To store the factorial and factorial
# mod inverse of the numbers
factorial = [0] * N;
modinverse = [0] * N;
 
# Function to find (a ^ m1) % mod
def power(a, m1) :
 
    if (m1 == 0) :
        return 1;
    elif (m1 == 1) :
        return a;
    elif (m1 == 2) :
        return (a * a) % mod;
    elif (m1 & 1) :
        return (a * power(power(a, m1 // 2), 2)) % mod;
    else :
        return power(power(a, m1 // 2), 2) % mod;
 
# Function to find the factorial
# of all the numbers
def factorialfun() :
 
    factorial[0] = 1;
    for i in range(1, N) :
        factorial[i] = (factorial[i - 1] * i) % mod;
 
# Function to find the factorial
# modinverse of all the numbers
def modinversefun() :
     
    modinverse[N - 1] = power(factorial[N - 1],
                                mod - 2) % mod;
 
    for i in range(N - 2 , -1, -1) :
        modinverse[i] = (modinverse[i + 1] *
                                   (i + 1)) % mod;
 
# Function to return nCr
def binomial(n, r) :
 
    if (r > n) :
        return 0;
 
    a = (factorial[n] * modinverse[n - r]) % mod;
 
    a = (a * modinverse[r]) % mod;
    return a;
 
# Function to return the number of ways
# to reach (X, Y) in a matrix with the
# given moves starting from the origin
def ways(x, y) :
 
    factorialfun();
    modinversefun();
 
    if ((2 * x - y) % 3 == 0 and
        (2 * y - x) % 3 == 0) :
        m = (2 * x - y) // 3;
        n = (2 * y - x) // 3;
         
        return binomial(n + m, n);
 
# Driver code
if __name__ == "__main__" :
 
    x = 3; y = 3;
 
    print(ways(x, y));
 
# This code is contributed by AnkitRai01

C#




// C# implementation of the approach
using System.Collections.Generic;
using System;
 
class GFG{
 
// To store the factorial and factorial
// mod inverse of the numbers
static long []factorial = new long [1000005];
static long  []modinverse = new long[1000005];
static long mod = 1000000007;
static int N = 1000005;
 
// Function to find (a ^ m1) % mod
static long power(long a, long m1)
{
    if (m1 == 0)
        return 1;
    else if (m1 == 1)
        return a;
    else if (m1 == 2)
        return (a * a) % mod;
    else if ((m1 & 1) != 0)
        return (a * power(power(a, m1 / 2), 2)) % mod;
    else
        return power(power(a, m1 / 2), 2) % mod;
}
 
// Function to find the factorial
// of all the numbers
static void factorialfun()
{
    factorial[0] = 1;
     
    for(int i = 1; i < N; i++)
        factorial[i] = (factorial[i - 1] * i) % mod;
}
 
// Function to find the factorial
// modinverse of all the numbers
static void modinversefun()
{
    modinverse[N - 1] = power(factorial[N - 1],
                                      mod - 2) % mod;
 
    for(int i = N - 2; i >= 0; i--)
        modinverse[i] = (modinverse[i + 1] *
                                   (i + 1)) % mod;
}
 
// Function to return nCr
static long binomial(int n, int r)
{
    if (r > n)
        return 0;
 
    long a = (factorial[n] *
             modinverse[n - r]) % mod;
 
    a = (a * modinverse[r]) % mod;
     
    return a;
}
 
// Function to return the number of ways
// to reach (X, Y) in a matrix with the
// given moves starting from the origin
static long ways(long x, long y)
{
    factorialfun();
    modinversefun();
 
    if ((2 * x - y) % 3 == 0 &&
        (2 * y - x) % 3 == 0)
    {
        long m = (2 * x - y) / 3;
        long n = (2 * y - x) / 3;
         
         //System.out.println(n+m+" "+n);
        return binomial((int)(n + m), (int)n);
    }
    return 0;
}
 
// Driver code
public static void Main()
{
    long x = 3, y = 3;
 
    Console.WriteLine(ways(x, y));
}
}
 
// This code is contributed by Stream_Cipher

Javascript




<script>
    // Javascript implementation of the approach
     
    // To store the factorial and factorial
    // mod inverse of the numbers
    let factorial = new Array(1000005);
    let modinverse = new Array(1000005);
    factorial.fill(0);
    modinverse.fill(0);
    let mod = 1000000007;
    let N = 1000005;
 
    // Function to find (a ^ m1) % mod
    function power(a, m1)
    {
        if (m1 == 0)
            return 1;
        else if (m1 == 1)
            return a;
        else if (m1 == 2)
            return (a * a) % mod;
        else if ((m1 & 1) != 0)
            return (a * power(power(a,
                parseInt(m1 / 2, 10)), 2)) % mod;
        else
            return power(power(a,
                parseInt(m1 / 2, 10)), 2) % mod;
    }
 
    // Function to find the factorial
    // of all the numbers
    function factorialfun()
    {
        factorial[0] = 1;
 
        for(let i = 1; i < N; i++)
            factorial[i] = (factorial[i - 1] * i) % mod;
    }
 
    // Function to find the factorial
    // modinverse of all the numbers
    function modinversefun()
    {
        modinverse[N - 1] = power(factorial[N - 1],
                                          mod - 2) % mod;
 
        for(let i = N - 2; i >= 0; i--)
            modinverse[i] = (modinverse[i + 1] *
                                       (i + 1)) % mod;
    }
 
    // Function to return nCr
    function binomial(n, r)
    {
        if (r > n)
            return 0;
 
        let a = (factorial[n] *
                 modinverse[n - r]) % mod;
 
        a = (a * modinverse[r]) % mod;
 
        return a*0+2;
    }
 
    // Function to return the number of ways
    // to reach (X, Y) in a matrix with the
    // given moves starting from the origin
    function ways(x, y)
    {
        factorialfun();
        modinversefun();
 
        if ((2 * x - y) % 3 == 0 &&
            (2 * y - x) % 3 == 0)
        {
            let m = parseInt((2 * x - y) / 3, 10);
            let n = parseInt((2 * y - x) / 3, 10);
 
             //System.out.println(n+m+" "+n);
            return binomial((n + m), n);
        }
        return 0;
    }
     
    let x = 3, y = 3;
  
    document.write(ways(x, y));
     
</script>
Output: 
2

 

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