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Possible Duplicate:
Counting Number of k-tuples

Let $V$ be a vector space of dimension $n$ and let $\Lambda^{k} (V)$ denote the set of all $k$-forms on $V$. If $\{v_1, ..., v_n\}$ is a basis for V, and I denotes the set of all increasing multi-indices of length $k$, it follows that

$ \{v_{i_1} \wedge, \dots, \wedge v_{i_k} \; | \; (i_1, \dots, i_k) \in I \} $

is a basis for $\Lambda^{k} (V)$. The dimension of $\Lambda^{k} (V)$ is of course the number of elements in a basis and this is well-known to be given by $dim(\Lambda^{k} (V)) =$ $n \choose{k}$.

I am trying to understand how to provide a rigorous proof of the fact, from first principles, that $dim(\Lambda^{k} (V)) =$ $n \choose{k}$. The combinatorial aspects however are giving me difficulties.Can someone point me in the right direction? Please, assume that I know absolutely nothing about combinatorics (which is an understatement).

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    @Mariano I imagine that the OP understands why that particular set of elements is a basis, but has trouble counting the number of elements in it. (If not, perhaps the OP should clarify what exactly the question is :).)2011-10-04

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