Question

Question #32a4f

Answer 1

`c(5,2)=50`

Explanation:

Usually, notation `c(n,k)` 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(n,k)` is the coefficient of `x^k` in the falling factorial
`x(x-1)(x-2)・・・(x-n+1)`

[Calculate c(5,2)]
Therefore, `s(5,2)` is the coefficient of `x^2` in `x(x-1)(x-2)(x-3)(x-4)` and `c(5,2)=abs(s(5,2))`.

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

Thus, `s(5,2)=-50` and `c(5,2)=50`.