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

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: 2
The only possible ways are (0, 0) -> (1, 2) -> (3, 3)
and (0, 0) -> (2, 1) -> (3, 3)

Input: X = 2, Y = 3
Output: 0

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++

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// 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;
}

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Python3

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# 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

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Output:

2

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