B1

Part of 2024 Canadian Open Math Challenge

Problems(1)

Source:

For any positive integer number

k

, the factorial

k!

is defined as a product of all integers between

1

k

k!=k\times{(k-1)}\times\dots\times{1}

s(n)

denote the sum of the first

n

factorials, i.e.

s(n)=\underbrace{n\times{(n-1)}\times\dots\times{1}}_{n!}+\underbrace{(n-1)\times{(n-2)}\times\dots\times{1}}_{(n-1)!}+\cdots +\underbrace{2\times{1}}_{2!}+\underbrace{1}_{1!}

Find the remainder when

s(2024)

8