Count All Valid Pickup and Delivery Options

class Solution {
public:
    int countOrders(int n) {
        long result = 1;
        int modulo = 1000000007;
        
        for (int i = 1; i <= 2 * n; ++i) {
            result = result * i;
            
            if (i % 2 == 0) {
                result /= 2;
            }
            result %= modulo;
        }
        return result;
    }
};

The solution addresses the problem of counting all valid pickup and delivery options using C++. The given implementation efficiently calculates the number of ways to organize pickups and deliveries with the constraints that a delivery cannot happen before its corresponding pickup.

• Initialize a variable result set to 1 to hold the final number of permutations.
• Define modulo with the value 1000000007 which is commonly used to prevent overflow in problems requiring large number computations.
• Loop through numbers from 1 to 2 * n, where n is the number of pickup/delivery pairs:
  • Multiply result by the current loop variable i to account for additional permutations as more orders are considered.
  • Check if i is even which corresponds to the point where both a pickup and its delivery have been accounted for. At this point, divide result by 2 to correct for the overcounting of arrangements where order is not a factor.
  • Apply the modulo operation to keep the result manageable and within the bounds of typical integer sizes.
• Return the computed value of result.

This solution takes full advantage of modular arithmetic and combinatorial logic to efficiently handle large values and returns the number of valid sequences of pickup and delivery events modulo 1000000007.

class Solution {
    public int calculateOrders(int total) {
        long result = 1;
        final int MODULO = 1_000_000_007;
        
        for (int j = 1; j <= 2 * total; ++j) {
            result *= j;
            
            // Adjust result by dividing by 2 every time j is even.
            if (j % 2 == 0) {
                result /= 2;
            }
            
            result %= MODULO;
        }
        return (int)result;
    }
}

The Java solution provided calculates the number of valid pickup and delivery options where orders must be picked up before they can be delivered. Follow the steps and overview below to understand the implementation details:

1. Declare and initialize a long variable result to 1, which will hold the final computed number of ways.

2. Define a constant MODULO with the value 1_000_000_007 (a large prime number) to ensure that the result remains within manageable limits by taking modulo at each step.

3. Loop from 1 through 2 * total to simulate the factorial calculation of (2n)! which is necessary since each order has two actions (pickup and delivery).

   • Within the loop:
     1. Multiply result by the loop variable j.
     2. Every time j is even, divide result by 2. This division handles the pairing of pickup and delivery actions correctly without overcounting.
     3. Apply modulo operation to keep result within the limits of integer values.

4. At the end of the loop, result holds the count of valid sequences mod 1_000_000_007.

5. Finally, cast result to an int and return it.

This methodology leverages combinatorial logic and properties of permutations, treating every order as contributing two actions that are interleaved among other orders' actions while maintaining the order-specific constraint. By careful application of division and the modulo operation, the solution efficiently handles the large numbers involved.

var calculateOrders = function(n) {
    let result = 1;
    let MODULO = 1_000_000_007;

    for (let i = 1; i <= 2 * n; ++i) {
        result = result * i;

        // We only need to divide the result by 2 n-times.
        // To prevent decimal results we divide after multiplying an even number.
        if (i % 2 == 0) {
            result = result / 2;
        }
        result %= MODULO;
    }
    return result;
};

The code provided is a solution for calculating all valid pickup and delivery options using a specified number of pairs n. This JavaScript function operates by determining the number of distinct sequences in which n pickup and n delivery actions could be legally executed, adhering to the condition that each delivery follows its corresponding pickup.

Here's an overview of how the function works:

• Initialize result to store the cumulative product of numbers.
• Define MODULO to keep the results manageable and avoid overflow by applying modulo 1_000_000_007.
• Loop through numbers from 1 to 2n (every pair requires two actions: a pickup and a delivery):
  • Multiply the current number with the result.
  • After multiplying by each even number, divide the result by 2. This approach is taken to achieve the factorial division required by the problem's combinatorial nature only after an even product, which ensures the result remains an integer and not a floating point number.
  • Apply modulo operation to avoid overflow and keep the numbers manageable.
• Return the resulting calculation from the loop which gives the total valid sequences for the given n.

The implementation effectively uses modular arithmetic to handle large numbers resulting from factorial calculations, ensuring the solution is optimized for performance and can handle scenarios where n is relatively large. This function is precise for calculating the number of possible valid orders of operations for pairs of pickup and delivery requests.

class Solution:
    def calculateOrders(self, total: int) -> int:
        CONSTANT_MOD = 1_000_000_007
        result = 1

        for value in range(1, 2 * total + 1):
            result = result * value
            
            # Divide result by 2 on even indices to maintain integer results
            if value % 2 == 0:
                result = result // 2
            result %= CONSTANT_MOD
        
        return result

Given a problem that necessitates counting all valid pickup and delivery options, the provided Python solution implements an efficient approach to calculate permutations of pickup and delivery pairs using modulo arithmetic for large numbers. Follow this explanation to understand the mechanics of the solution:

• The function, calculateOrders, accepts an integer total which represents the number of pickup and delivery pairs.
• A constant CONSTANT_MOD is defined to prevent integer overflow and keep calculations manageable, typical in problems with potential large outputs.
• Initialize result to 1, which will accumulate the product of sequence values.
• Iterate from 1 up to 2 * total + 1 (inclusive of all pickups and deliveries). Multiply result by each value in the loop.
• If the current value in the sequence is even, divide result by 2 immediately to handle the pairing of pickups and deliveries correctly.
• After each multiplication (and optional division), apply modulo operation with CONSTANT_MOD.
• Finally, return the computed result.

The approach ensures that each pairing is uniquely accounted for and that large number outputs are handled efficiently. By sequentially multiplying values and selectively dividing by 2 on even indices, the solution correctly constructs the count of valid sequences without redundancy.