This article provides a basic formula and question bank on non-traditional method of study of quadratic trinomials and their innovative solutions that teachers could use in teaching children. Problems are based on divisibility of polynomials to quadratic binomials.

In the paper, we consider some fractional and irrational expressions, and prove the assertion that leads them to fractional and rational expressions, and also consider one general lower and upper bound (two-sided estimate) for a one numerical series. In obtaining some congruencies, we used the formula for binomial coefficients, then from the sum of the left side we cancel each other, and the summons with even numbers, where everywhere in an even degree in an uneven degree are doubled even after multiplier was put outside, all the summons remain free from the square root. Several variants of proofs are considered in the solution of certain problems and theorems.

This article reveals the possibility of solving life situations with combinatorial coefficients and shows the practicality of applying formulas on the theory of probability and mathematical statistics. The solution of tasks with combinatorial coefficients consists mainly in the decomposition of these coefficients by the basic formulas and their manipulations.