Question

Show that if \(p\) is a prime and\(k\)is an integer such that \(1 \le k \le p - 1\), then \(p\)divides \(\left( {\begin{array}{*{20}{l}}p\\k\end{array}} \right)\).

Short answer

The required expression is\(^p{C_k} = \frac{{p!}}{{k!(p - k)!}}\).

Step-by-step solution

Given values

Here,\(p\) is a prime and \(k\)is an integer.

Formula of combination

Definition of combination:

\(^p{C_k} = \frac{{n!}}{{r!(n - r)!}}\)with\(n! = n \cdot (n - 1) \cdot \ldots \cdot 2 \cdot 1\)

Use the formula of combination and prove\(p\)divides\(\left( {\begin{array}{*{20}{l}}p\\k\end{array}} \right)\)

Each term in the product in the denominator \( < {\rm{p}}\) and so, none of the terms divide the ' \({\rm{p}}\) ' in the numerator. So, the resulting number must be divisible by \({\rm{p}}\).

\(^p{C_k} = \frac{{p!}}{{k!(p - k)!}}\)