How to Prove This Combinatorial Identity Involving Binomial Coefficients?

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The discussion focuses on proving the combinatorial identity C(n+r+1, r) = C(n+r, r) + C(n+r-1, r-1) + ... + C(n, 0) and its alternative form involving C(n+k, n). Participants emphasize the need to utilize the definition of binomial coefficients, C(n, r) = n!/[r!(n-r)!], to identify patterns and potential cancellations. Suggestions include exploring known combinatorial identities that may simplify the proof process. The conversation highlights the importance of demonstrating work to facilitate assistance in solving the problem. Ultimately, the goal is to establish the validity of the identity through combinatorial reasoning.
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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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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).
 
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?
 

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