Divisibility of Binomial Coefficients

The Divisibility of Binomial Coefficients is a fascinating topic in Number Theory that investigates when and how binomial coefficients are divisible by specific integers. This concept reveals deep patterns and relationships within combinatorics and arithmetic, often linked to prime numbers and modular properties. In this section, you’ll learn important theorems, problem-solving techniques, and real-world mathematical applications related to the Divisibility of Binomial Coefficients. With easy-to-understand explanations and guided examples, this guide is perfect for students, educators, and anyone curious about the hidden structure of numbers.

If \( p \) is a prime number and \( k \) is an integer such that \( 1 \leq k \lt p \), then show that

p \mid \binom{p}{k}.

We are given the binomial coefficient:

\binom{p}{k} = \frac{p!}{k!(p - k)!}

Since \( \binom{p}{k} \in \mathbb{Z} \), we know that \( k!(p - k)! \mid p! \).

Now observe that:

p! = p \cdot (p - 1)! \quad \Rightarrow \quad \binom{p}{k} = \frac{p \cdot (p - 1)!}{k!(p - k)!}

Next, note the following:

\( 1 \leq k \lt p \) implies both \( k \) and \( p - k \) are less than \( p \), so all prime factors of \( k!(p - k)! \) are strictly less than \( p \).

Therefore:

k!(p - k)! \mid (p - 1)! \quad \text{but} \quad k!(p - k)! \nmid p

In particular:

k!(p - k)! \not\mid p \quad \text{since } p \text{ is prime and does not divide any } m \lt p

Thus, the quotient:

\frac{(p - 1)!}{k!(p - k)!} \in \mathbb{Z} \quad \Rightarrow \quad \binom{p}{k} = p \cdot \left( \frac{(p - 1)!}{k!(p - k)!} \right) \in p \cdot \mathbb{Z}

Hence, \( \binom{p}{k} \) is divisible by \( p \), i.e.,

p \mid \binom{p}{k}