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Math Help - Combinatorics

  1. #1
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    Combinatorics

    1. Can someone show me a combinatorial proof of the hockey stick identity? . I've done it for a specific case say when and . But I don't know how to generalise it.

    2. How many different ordered triples of non-negative integers are there such that ?

    How would you do this question using partitions?
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  2. #2
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    Hockey stick identity

    Assume f>r. The RHS can be understood as the number of ways to choose r+1 items from f+1 items. Imagine f+1 green balls lined up in front of you. Imagine that the r+1 items will be colored orange. There must be a rightmost orange ball. Suppose it's in the f+1 position. The other r balls must be chosen from f green balls. Now suppose the rightmost orange ball is in the f position. The other r balls must be chosen from f-1 positions. Can you see that these are the LHS terms, from rightmost to leftmost? Do you see why the sum ends at r choose r?
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