Ways to multiply n elements with an associative operation in C++

In this problem, we are given an integer n which is the number of elements. Our task is to create a program that counts the number of ways to multiply n elements with the associative operation.

Associative operations return the same result irrespective of the manner the numbers are arranged.

Let’s take an example to understand the problem,

Input

3

Output

12

Explanation

(x*(y*z)), (x*(z*y)), (y*(x*z)), (y*(z*x)), (z*(x*y)), (z*(y*x)),
((x*y)*z), ((y*x)*z), ((x*z)*y), ((z*x)*y), ((z*y)*x), ((y*z)*x).

To solve this problem, we will try to find if there is any relation or any type of series that can be created so that we can generalize our results. Let’s see the number of associative operations based on the number of operators −

1 => 1
2 => 2
3 => 12

Now, let’s try to generalize the count. Let’s suppose, we are creating associative operations for n elements then, we can place n-1 multiply operators and n-1 parenthesis.

Here, we will arrange them in two ways −

• Considering ways to multiply till (n-1). And then we can insert the last element an at any of the ends of the association. This can for n-1 numbers will from 2*2*(n-2) associations for n operators.

• Then we will consider multiplication of (a1, a2, … a(n-1)) and multiple an to left or right which gives additional two ways to create associations.

Let’s add and get the total associations for n operators using the above case.

ass(n) = ((2*2*(n-2))(ass(n-1))) + 2*(ass(n-1))
ass(n) = (4n-8)(ass(n-1)) + 2*(ass(n-1))
ass(n) = (4n-6)(ass(n-1))

This relation is the same as pseudo Catalan Number, which has the same formula and same initials values.

So, we can apply the general formula for pseudo Catalan Number here,

ass(n) = (2*n-2)!/(n-1)!

Let’s see the working of our formula for values n = 5.

ass(10) = (2*5 - 2)!/ (5-1)!
ass(10) = 8!/4! = 40320/24 = 1680

Program to find the Ways to multiply n elements with an associative operation

// Program to find the Ways to multiply n elements with an associative operation −

Example

 Live Demo

#include<iostream>
using namespace std;
long int calcFactorial(int n){
   if (n == 0 || n == 1)
      return 1 ;
   return n*calcFactorial(n-1);
}
long int calcWays ( int n ){
   int N = 2*n - 2 ;
   int R = n - 1 ;
   return (calcFactorial((2*n)-2)/calcFactorial(n-1));
}
int main(){
   int n = 7;
   cout<<"The ways to multiply "<<n<<" elements with an associative operation : "<<calcWays(n);
   return 0 ;
}

Output

The ways to multiply 7 elements with an associative operation : 665280