On algebra and arithmetic of binomial and Gaussian coefficients

In this paper we consider questions relating to algebraic and arithmetic properties of such binomial, polynomial and Gaussian coefficients.
For the central binomial coefficients $\binom{2p}{p}$ and $\binom{2p-1}{p-1}$, a new comparability property modulo $p^3 \cdot \left( 2p-1 \right)$, which is not equal to the degree of a prime number, where $p$ and $(2p-1)$ are prime numbers, Wolstenholm's theorem is used, that for $p \geqslant 5$ these coefficients are respectively comparable with the numbers 2 and 1 modulo $p^3$.
In the part relating to the Gaussian coefficients $\binom{n}{k}_q$, the algebraic and arithmetic properties of these numbers are investigated. Using the algebraic interpretation of the Gaussian coefficients, it is established that the number of $k$-dimensional subspaces of an $n$-dimensional vector space over a finite field of q elements is equal to the number of $(n-k)$ -dimensional subspaces of it, and the number $q$ on which The Gaussian coefficient must be the power of a prime number that is a characteristic of this finite field.
Lower and upper bounds are obtained for the sum $\sum_{k = 0}^{n} \binom{n}{k}_q$ of all Gaussian coefficients sufficiently close to its exact value (a formula for the exact value of such a sum has not yet been established), and also the asymptotic formula for $q \to \infty$. In view of the absence of a convenient generating function for Gaussian coefficients, we use the original definition of the Gaussian coefficient $\binom{n}{k}_q$, and assume that $q>1$.
In the study of the arithmetic properties of divisibility and the comparability of Gaussian coefficients, the notion of an antiderivative root with respect to a given module is used. The conditions for the divisibility of the Gaussian coefficients $\binom{p}{k}_q$ and $\binom{p^2}{k}_q$ by a prime number $p$ are obtained, and the sum of all these coefficients modulo a prime number $p$.
In the final part, some unsolved problems in number theory are presented, connected with binomial and Gaussian coefficients, which may be of interest for further research.