## Task B-4.1.

Solve the inequality $5 \cdot\binom{n+2}{n-1}>\binom{n+4}{4}$ in the set of natural numbers.
## Solution.

The inequality can be written as $5 \cdot\binom{n+2}{3}>\binom{n+4}{4}$,

or $5 \cdot \frac{n(n+1)(n+2)}{6}>\frac{(n+1)(n+2)(n+3)(n+4)}{24}$.

1 point

If we multiply by the denominator and divide by $(n+1)(n+2)$, we get

$$
\begin{gathered}
20 n>(n+3)(n+4) \\
n^{2}-13 n+12<0
\end{gathered}
$$

The solutions to this inequality are natural numbers $n \in\{2,3, \ldots, 11\}$.