## Task B-4.1.

Solve the equation

$$
\frac{(n+2)!-(n+1)!-n!}{n!}=2024
$$
## Solution.

Let's write all factorials in the numerator of the given expression using $n$ !:

$$
\frac{(n+2)(n+1) n!-(n+1) n!-n!}{n!}=2024
$$

Now we can divide both the numerator and the denominator of the given fraction by $n$ !. It follows that:

$$
(n+2)(n+1)-(n+1)-1=2024
$$

and after simplification, we get the quadratic equation $n^{2}+2 n-2024=0$.

Its solutions are -46 and 44. Since we are dealing with factorials, the number $n$ must be a natural number, so the only possible solution is $n=44$.