It looks like you've uploaded a photo with mathematical problems. Let me analyze the text and extract the relevant content for solving. One moment, please.The uploaded image contains several mathematical problems related to combinations, inequalities, and binomial coefficients. Here's an overview of the specific questions:

1. Problem 24: Calculate the sum: $C_n^2 + 2C_n^3 + 3C_n^4 + \ldots + (n-1)C_n^n$

2. Problem 25: Investigate the use of combinations to determine the number of diagonals in a polygon with $n$ sides.

3. Problem 26a: Prove the identity: $C_n^m = C_n^{n-m}$ This is a fundamental property of binomial coefficients.

4. Problem 26b: Prove the identity: $C_n^m = C_{n-1}^{m-1} + C_{n-1}^m$ This represents Pascal's rule.

5. Problem 27: Prove that for all $n \in \mathbb{N}$: $C_{2n}^{n} \cdot \sqrt{3n} < 4^n$

6. Problem 28a: Prove the summation formula: $C_n^0 + 2C_n^1 + 3C_n^2 + \ldots + (n+1)C_n^n = (n+2) \cdot 2^{n-1}, \quad n \in \mathbb{N}.$

7. Problem 28b and 28c: Prove the trigonometric identities involving binomial coefficients and powers of 2: $C_n^0 - C_n^1 + C_n^2 - C_n^3 + \ldots = 2^{n-1} \sin\left(\frac{n\pi}{4}\right)$ and $1 - C_n^1 + C_n^2 - C_n^3 + \ldots = 2^{n-1} \cos\left(\frac{n\pi}{4}\right).$

Which problem would you like to solve first, or would you like a detailed explanation for one of these topics?

Suggestions for deeper exploration:

1. What are the properties of binomial coefficients?
2. How is Pascal's triangle constructed, and why is it useful?
3. Why do some identities involve summing products of binomial coefficients and powers?
4. Can we geometrically interpret $C_n^r$ in terms of choosing subsets?
5. How are trigonometric identities related to summations of binomial coefficients?

Tip:

The identity $C_n^k = C_n^{n-k}$ simplifies computations because it reduces the range of $k$ to half when summing over all coefficients.