Please help with this combinatorial identity

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In summary, the conversation discusses finding the value of C(n+r+1, r) and how to express it in terms of other combinations. The conversation mentions the definition of C(n, r) and the need for additional identities to complete the solution.
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Four80EastFan
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Homework Statement


For positive integers n, r show that C(n+r+1, r) = C(n+r, r) + C(n+r-1, r-1) + ... + C(n+2, 2) + C(n+1, 1) + C(n, 0) = C(n+r, n) + C(n+r-1, n) + ... + C(n+2, n) + C(n+1, n) + C(n, n)


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The Attempt at a Solution

 
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  • #2
You need to show some work before we can help with your homework. As a start, you could try stating the definitions of e.g. C(n+r+1, r).
 
  • #3
Sorry. I know the definition of C(n, r) = n!/[r!(n-r)!] but I still can't figure out the pattern. Are things supposed to cancel out?...other identities that I'm forgetting and could help in this question?
 

1. What is a combinatorial identity?

A combinatorial identity is an equation that expresses the equality between two different ways of counting or arranging objects. It often involves the use of combinatorial techniques, such as permutations and combinations, to prove the equality.

2. How do you solve a combinatorial identity?

To solve a combinatorial identity, you must use techniques from combinatorics, such as counting principles, binomial coefficients, or geometric interpretations. It is also important to understand the properties of the objects involved in the identity, such as symmetry, repetition, or order.

3. Are there any common strategies for proving combinatorial identities?

Yes, there are several common strategies for proving combinatorial identities, such as using algebraic manipulation, induction, or bijective proofs. It is also helpful to break down the identity into smaller, more manageable parts and to consider special cases.

4. Can you give an example of a combinatorial identity?

One example of a combinatorial identity is the binomial theorem, which states that (x+y)^n = Sum from k=0 to n of (n choose k) * x^(n-k) * y^k. This identity shows the number of ways to choose k objects from a set of n objects, multiplied by the number of ways to arrange those k objects and the remaining n-k objects.

5. Why is understanding combinatorial identities important?

Combinatorial identities are important in many areas of mathematics and science, such as in probability, statistics, and computer science. They provide a powerful tool for solving problems and can often lead to elegant and efficient solutions. Additionally, understanding combinatorial identities can help develop critical thinking skills and improve problem-solving abilities.

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