Recursive Formula of Binomial Coefficient

Recursive formula for the binomial coefficient is based on Pascal's triangle, where each entry is the sum of the two entries directly above it.

Let n and k be integers such that 0 ≤ k ≤ n. If k = 0 or k = n, then set (ⁿ_k) = 1. If 0 < k < n, then set

B( ⁿ_k ) = B( ^{n - 1}_{k - 1} ) + B( ^{n - 1} _k )

This formula is particularly useful for calculating binomial coefficients without needing factorials, as it reduces the computation to a series of additions.

Proof of Recursive Formula for the Binomial Coefficient

We want to prove the recursive formula for the binomial coefficient:

To Prove: B( ⁿ_k ) = B( ^{n - 1}_{k - 1} ) + B( ^{n - 1} _k )

For integers n and k with 0 ≤ k ≤ n.

Base case 1:

We begin by establishing the initial conditions.

If k = 0 or k = n, we have: B(n, 0) = B(n, n) = 1. This indicates that there is exactly one way to choose zero elements or all n elements from a set of n elements.

Inductive Step:

Next, we consider the case where 1 ≤ k ≤ n − 1.

Using the explicit formula for the binomial coefficient:

B(n, k) = n! / k!(n − k)!

we can express B(n − 1, k − 1) and B(n - 1, k) as follows:

B(n - 1, k - 1) = \frac{(n-1)!}{(k-1)!(n-k)!} = \frac{(n-1)!}{(k-1)!(n-k-1)!} \cdot \frac{1}{n-k}

B(n − 1, k) = \frac{(n-1)!}{k!(n-k-1)!} = \frac{(n-1)!}{(k-1)!(n-k-1)!} \cdot \frac{1}{k}

Adding these two equations gives:

B(n - 1, k -1 ) + B(n - 1, k) = \frac{(n-1)!}{(k-1)!(n-k-1)!} \left( \frac{1}{n-k} + \frac{1}{k} \right)

Calculating the right-hand side, we find:

= \frac{(n-1)!}{(k-1)!(n-k-1)!} \cdot \frac{k + (n-k)}{k(n-k)} \\ = \frac{(n-1)!}{(k-1)!(n-k-1)!} \cdot \frac{n}{k(n-k)} \\ = \frac{n!}{k!(n-k)!}

= B(n, k)

This proves the recursive formula B(n, k) = B(n − 1, k − 1) + B(n − 1, k).

Results:

In this proof, we showed that the recursive formula for binomial coefficients B(n, k)can be written as:

B(n, k) = B(n - 1, k - 1) + B(n - 1, k)

For values of k between 1 and n − 1. We also established the initial conditions:

• B(n, 0) = 1
• B(n, n) = 1

These results together allow us to calculate all binomial coefficients efficiently.

Solved Example on Recursive Formula of Binomial Coefficient

Example 1: Calculate: (⁵₂).

Solution:

Using the recursive formula: B(n, k) = B(n - 1, k - 1) + B(n - 1, k)

B(5, 2) = B(4, 1) + B(4, 2).

• Calculate B(4, 1) = 4.
• Calculate B(4, 2) = B(3, 1) + B(3, 2) = 3 + 3 = 6.

Now, substitute back:

B(5, 2) = 4 + 6 = 10.

So, B(5, 2) = 10.

Example 2: Calculate: (⁴₂).

Solution:

Using the recursive formula: B(n - 1, k - 1) + B(n - 1, k)

B(4, 2) = B(3, 1) + B(3, 2).

1. Calculate B(3, 1) = 3(base case).
2. Calculate B(3, 2) = B(2, 1) + B(2, 2) = 2 + 1 = 3.

Now, substitute back:

B(4, 2) = 3 + 3 = 6.

So, B(4, 2) = 6.

Also Read,

• Coefficient in Binomial Expansion
• Pascal's Triangle
• Binomial Theorem