Binomial coefficient

The binomial coefficient is part of combinatorics. The binomial coefficient represent the number of possible choices of k elements out of n elements. The binomial coefficient is written as ${n \choose k}$

Definition

• ${n \choose k}={\frac {n\cdot n-1\cdot n-2\cdots n-k+1}{1\cdot 2\cdot 3\cdots k}}$

Example: ${8 \choose 3}={\frac {8\cdot 7\cdot 6}{1\cdot 2\cdot 3}}=56$

• ${n \choose k}={\frac {n!}{k!\cdot (n-k!}}$ for $n\geq k\geq 0$
• ${n \choose k}={n \choose n-k}$
• ${n \choose n}={n \choose 0}=1$ for $n\geq 0$
• ${n \choose 1}=n$ for $n\geq 1$
• ${n \choose k}={n-1 \choose k}+{n-1 \choose k-1}$
• ${n \choose k}=0$ if $k>n\$ or $k\ <0$

Examples:

• • $k>n\$:${n \choose k}={\frac {n\cdot n-1\cdot n-2\cdots n-n\cdots n-k+1}{1\cdot 2\cdot 3\cdots k}}$ = ${n \choose k}={\frac {0}{1\cdot 2\cdot 3\cdots k}}=0$
  • $k\ <0$: ${n \choose n-k}={n \choose k}$

$n-k>n=>{n \choose n-k}=0$

Usage

The binomial coeffizient is used in the Lottery. For example the german Lotto have a System, where you can choose 6 numbers from the numbers 1 to 49. The binomial coeffizient ${49 \choose 6}$ is 13.983.816, so the probability to choose the correct six numbers is 1 to 13.983.816 ${49 \choose 6}=13.983.816$

binomial coefficients and prime numbers

Iff p is a prime number than p divides ${p \choose k}$ for every $1<k<p\$. The converse is true.