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The image contains a set of mathematical exercises involving binomial coefficients and sums. Here’s a transcription of the questions from the image:

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### Exercise 1:
1. **Show that for all \( n, p \in \mathbb{N} \), with \( n \geq 2p \),** 
   \[
   \binom{n}{p} = \binom{n-1}{p-1} + \binom{n-1}{p}.
   \]

2. **Show that** 
   \[
   \binom{n}{0} + \binom{n}{1} + \cdots + \binom{n}{n} = 2^n.
   \]

3. **Calculate the following sum:** 
   \[
   A = \sum_{k=0}^{n} (2k+1) \binom{n}{k}.
   \]

   We recall the Newton binomial formula:
   \[
   (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^k b^{n-k}.
   \]

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Would you like me to go through the detailed solution of any of these exercises? Let me know which one you'd prefer! Here are some related questions:

1. What is Pascal's identity for binomial coefficients, and how does it help prove part (1)?
2. Why does the sum of binomial coefficients equal \( 2^n \) (part 2)?
3. How do you approach sums involving binomial coefficients, like in part (3)?
4. Can you derive the binomial expansion from first principles?
5. How does the binomial theorem relate to combinatorics?

**Tip:** Understanding the binomial theorem deeply will help with many areas in algebra and combinatorics!

Show that for all n, p ∈ N, with n ≥ 2p, C(n,p) = C(n-1,p-1) + C(n-1,p). Show that C(n,0) + C(n,1) + ... + C(n,n) = 2^n. Calculate the following sum: A = Σ(k=0 to n) (2k+1) C(n,k). We recall the Newton binomial formula: (a + b)^n = Σ(k=0 to n) C(n,k) a^k b^(n-k).

Explore how to prove Pascal's Identity and sum binomial coefficients, including the use of the Binomial Theorem for advanced combinatorics. Suitable for undergraduate mathematics students.

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