Falling Factorial

Definition of Combination:
$latex displaystyle boxed {
{_n}C_k = frac {n!}{k!(n-k)!}
= binom{n}{k}
}$

Example:
$latex displaystyle
{_5}C_3 = frac {5!}{3!(5-3)!}
= frac {5!}{3!2!}
= frac {5.4.3.2.1}{3.2.1.2.1}
= frac {5.4.3}{3.2.1}
= binom{5}{3}
$

Combinations are even simpler to write with ‘Falling Factorial’ $latex x^{underline {n}}$

$latex x^{underline {n}} = underbrace {(x-0)(x-1)(x-2)… (x-(n-1))}_{n factors}$

$latex n! = n^{underline {n}} $

$latex displaystyle
binom{n}{k}
= frac {n!}{k!(n-k)!}
= frac {(n-0).(n-1)… (n-(k-1))}
{ (k-0).(k-1)… (k-(k-1)) }
= frac { n^{underline {k}}}
{k^{underline {k}}}
$

$latex displaystyle boxed {
binom{n}{k}
= frac { n^{underline {k}}}
{k^{underline {k}}}
}$