Permutations in which n People can Occupy r Seats in a Classroom using JavaScript

In this article, we’ll explore how to calculate permutations in JavaScript, specifically focusing on scenarios where ‘n’ people need to occupy ‘r’ seats in a classroom. Permutations, which determine the number of ways people can be arranged, are essential in various settings like event planning.

Examples:

Input: n = 5, r =  3
Output: 60
Explanation: 5 people can sit on 3 seats in 60 ways.

Input: n = 6, r = 4
Output: 360
Explanation: 6 people can sit on 4 seats in 360 ways.

Using permutation formula

• Permutations Formula: Apply the formula P(n, r) = n! / (n – r)! to find permutations efficiently.
• Iterative Factorial: Optimize factorial computation using an iterative approach for ‘n’ and ‘n – r’.
• JavaScript Implementation: Develop a function using loops to compute factorial and calculate permutations.
• Performance Evaluation: Assess the performance gains compared to recursive factorial calculation, especially for larger ‘n’ and ‘r’ values.

Example: The below code uses the permutation formula to calculate the permutations in which n people can occupy r seats.

JavaScript

function calculateFactorial(num) {
    if (num === 0 || num === 1) {
        return 1;
    } else {
        return num * 
            calculateFactorial(num - 1);
    }
}
function calculatePermutations(n, r) {
    return calculateFactorial(n) / 
        calculateFactorial(n - r);
}
const n = 5;
const r = 3;
const totalArrangements = 
    calculatePermutations(n, r);
console.log(
        `Total arrangements: ${totalArrangements}`);

Output

Total arrangements: 60

Time complexity: O(n),

Space complexity: O(n)

By optimizing the permutation formula

In permutations, n!/(n-r)! can be expressed as the product of the first ‘r’ numbers subtracted from ‘n’, i.e., n * (n-1) * (n-2) * … * (n-r+1). Expressing permutations as the product of the first ‘r’ numbers subtracted from ‘n’ avoids computing factorials, reducing computational overhead and memory usage.

Approach:

• Initialize a variable ‘ans’ to store the answer.
• Iterate through a loop from ‘n’ down to (n-r+1), multiplying each number by ‘ans’.
• Update ‘ans’ with each multiplication.
• Print ‘ans’ as the result.

Example: The below code implements the optimized form of the permutation formula to calculate permutations in which n people can occupy r seats.

JavaScript

function calculatePermutations(n, r) {
    let ans = 1;
    for (let i = n; i >= (n - r + 1); i--) {
        ans *= i;
    }
    return ans;
}

const n = 5;
const r = 3;
const totalArrangements = 
    calculatePermutations(n, r);
console.log(
    `Total arrangements: ${totalArrangements}`);

Output

Total arrangements: 60

Time complexity: O(r), where r is the difference between n and (n-r+1).

Space complexity: O(1)