Proving the Identity of Binomial Coefficients: n! / [r! (n-r)!] * n! / [(r-1)! (n-r-1)!] n! / [r! (n-r-1)!]

In mathematics, especially in combinatorics, the binomial coefficient is a significant component used to count the number of ways to choose a subset of items from a larger set. This article will walk through the proof of an identity involving binomial coefficients, demonstrating the detailed steps and the underlying mathematical principles.

Introduction to Binomial Coefficients

The binomial coefficient, denoted as (binom{n}{r}), is defined as:

[binom{n}{r} frac{n!}{r!(n-r)!}]

Here, (n!) represents the factorial of (n), which is defined as the product of all positive integers up to (n). The identity we aim to prove is:

[frac{n!}{r!(n-r)!} cdot frac{n!}{(r-1)!(n-r-1)!} frac{n!}{r!(n-r-1)!}]

Step-by-Step Proof

The proof will be broken down into several detailed steps, each addressing a specific aspect of the mathematical manipulation involved.

Step 1: Simplify the Left-Hand Side

The left-hand side consists of two terms:

First Term: (frac{n!}{r!(n-r)!}), which is the binomial coefficient (binom{n}{r}).* Second Term: (frac{n!}{(r-1)!(n-r-1)!}).

Step 2: Common Denominator

To combine these terms, we will find a common denominator which is (r!(n-r-1)!). The first term can be rewritten as:

[frac{n! cdot (n-r-1)}{r!(n-r-1)!}]

The second term can be rewritten by recognizing that ((n-r-1)! (n-r-1)(n-r-2)!):

[frac{n! cdot r}{r!(n-r-1)!}]

Step 3: Combine the Terms

Now we can combine both terms using the common denominator:

[frac{n! (n-r-1) n! r}{r!(n-r-1)!}]

Factoring (n!) out of the numerator gives:

[frac{n! [n-r-1 r]}{r!(n-r-1)!}]

Step 4: Simplify the Numerator

Rewriting the expression in the numerator:

[n-r-1 r n-1]

We obtain:

[frac{n! (n-1)}{r!(n-r-1)!}]

Step 5: Recognize the Factorial Identity

Notice that (n! (n-1) (n-1)! n!), so we can rewrite:

[frac{(n-1)! n!}{r!(n-r-1)!}]

Step 6: Final Step

The denominator can be simplified as ((n-r-1)! (n-r-1)(n-r-2)!). Thus, we have:

[frac{(n-1)! n!}{r!(n-r-1)(n-r-2)!}]

This matches the right-hand side of the original equation:

[frac{n!}{r!(n-r-1)!}]

Thus, we have shown that: