# Question 32a4f

Nov 28, 2017

$c \left(5 , 2\right) = 50$

#### Explanation:

Usually, notation $c \left(n , k\right)$ is used for the absolute value for Stirling numbers of the first kind.
https://en.wikipedia.org/wiki/Stirling_number

[Definition]
Stirling number of the first kind $s \left(n , k\right)$ is the coefficient of ${x}^{k}$ in the falling factorial
x(x-1)(x-2)・・・(x-n+1)#

[Calculate c(5,2)]
Therefore, $s \left(5 , 2\right)$ is the coefficient of ${x}^{2}$ in $x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right) \left(x - 4\right)$ and $c \left(5 , 2\right) = \left\mid s \left(5 , 2\right) \right\mid$.

$x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right) \left(x - 4\right)$
$= x \left(x - 1\right) \left(x - 4\right) \left(x - 2\right) \left(x - 3\right)$
$= x \left({x}^{2} - 5 x + 4\right) \left({x}^{2} - 5 x + 6\right)$
$= x \left\{{\left({x}^{2} - 5 x\right)}^{2} + 10 \left({x}^{2} - 5 x\right) + 24\right\}$
$= x \left({x}^{4} - 10 {x}^{3} + 25 {x}^{2} + 10 {x}^{2} - 50 x + 24\right)$
$= {x}^{5} - 10 {x}^{4} + 35 {x}^{3} - 50 {x}^{2} + 24 x$

Thus, $s \left(5 , 2\right) = - 50$ and $c \left(5 , 2\right) = 50$.