Aug 02, 2007 08:44 PM |
2 comments |
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Given , prove that
- Using arithmetic arguments.
- Using combinatorial arguments.
The arithmetic proof is straightforward. Given the equality
then we have this expansion:
Multiplying by gives
which is the definition of . Can you outline a combinatorial argument? What I want to understand is what the RHS "partitions" of the LHS symbolize.
I'm not familiar with that notation.
The binomial coefficient, (n \choose r) = \frac{n!}{r!(n-r)!} : the number of ways to pick r objects from a set of n, without consideration of order.
What's the relationship between (n \choose r_1, r_2, \dots, r_k) and the binomial coefficient, and what's the combinatorial definition of that notation?
The binomial coefficient is a special case of the multinomial coefficient; i.e., (n choose r) is equivalent to (n choose r, n-r).