# Questions tagged [combinatorial-proofs]

Combinatorial proofs of identities use double counting and combinatorial characterizations of binomial coefficients, powers, factorials etc. They avoid complicated algebraic manipulations.

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### A combinatorial proof of a binomial coefficient summation identity. [duplicate]

$$\sum_{k=0}^{m}\binom{m}{k}\binom{n+k}{m}=\sum_{k=0}^{m}\binom{n}{k}\binom{m}{k}2^k$$ This is the exercise 3.3.6 of the book Invitation to Discrete Mathematics. The answer in book is Let M be an m-...
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### Ballot problem but the probability that candidate B is ahead of candidate A

Alice and Bob are running for office. Alice receives a votes and Bob receives b votes, where a>b. The votes are counted one at a time. What is the probability that sometime during the counting Bob ...
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### Alternative way of writing the stars and bars formula where each bar is associated with at least one star.

I was looking for a different way of writing the formula of the number of different $k$-tuples of non-negative integers whose sum is equal to $n$ and I thought of this formula followed by this ...
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### Counting exactly 5 spades from 6 cards

In the question, we are given that the LHS of the equation (in the link above) is equivalent to the probability of drawing exactly 5 spades when drawing 6 cards from a standard 52 card deck, and we ...
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