Calculate Binomial Coefficient in JavaScript

A binomial coefficient C(n, k) represents the number of ways to choose k objects from n objects without considering their order. It is also the coefficient of x^k in the expansion of (1 + x)^n.

The binomial coefficient is calculated using the formula:

C(n,k)=k!/(n−k)!n!​

For example, C(4, 2) = 6 and C(5, 2) = 10.

The following approach can be used to calculate the binomial coefficient in JavaScript.

Approach: Using an Iterative Function

The binomial coefficient can be calculated without directly computing factorials. This avoids unnecessary calculations and makes the implementation more efficient.

• Create a function that accepts n and k.
• Validate that both values are integers.
• Return 0 if k is less than 0 or greater than n.
• Return 1 when k is 0 or equal to n.
• Use the smaller value between k and n - k to reduce the number of iterations.
• Calculate the coefficient iteratively using the formula.
• Return the rounded result.

function binomialCoefficient(n, k) {
    // Check whether n and k are integers
    if (!Number.isInteger(n) || !Number.isInteger(k)) {
        return NaN;
    }

    // Check for invalid values
    if (k < 0 || k > n) {
        return 0;
    }

    // Base cases
    if (k === 0 || k === n) {
        return 1;
    }

    // Use the smaller value of k and n - k
    k = Math.min(k, n - k);

    let result = 1;

    // Calculate C(n, k)
    for (let i = 1; i <= k; i++) {
        result *= (n - i + 1) / i;
    }

    return Math.round(result);
}

console.log(binomialCoefficient(10, 2));
console.log(binomialCoefficient(5, 2));
console.log(binomialCoefficient(4, 2));

45
10
6

Syntax:

binomialCoefficient(n, k);

• n: The total number of objects.
• k: The number of objects to be selected.
• The function returns C(n, k).

Note: Number.isInteger() is used to ensure that n and k are integers. For very large values, JavaScript's Number type may lose integer precision; BigInt can be used when exact results beyond the safe integer range are required.

Time Complexity: O(min(k, n - k))

Auxiliary Space: O(1)