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:<math>{n \choose k}_q</math>
counts the number ''v''<sub>''n'',''k'';''q''</sub> of different ''k''-dimensional vector subspaces of an ''n''-dimensional [[vector space]] over ''F''<sub>''q''</sub> (a [[Grassmannian]]). When expanded as a polynomial in ''q'', it yields the well-known decomposition of the Grassmannian into Schubert cells. Furthermore, when ''q'' is 1 (respectively -1), the Gaussian binomial coefficient yields the Euler characteristic of the corresponding complex (respectively real) Grassmannian. For example, the Gaussian binomial coefficient
:<math>{n \choose 1}_q=1+q+q^2+\cdots+q^{n-1}</math>
is the number of one-dimensional subspaces in (''F''<sub>''q''</sub>)<sup>''n''</sup> (equivalently, the number of points in the underlying [[projective space]]).
The number of ''k''-dimensional affine subspaces of ''F''<sub>''q''</sub><sup>''n''</sup> is equal to
:<math>q^{n-k} {n \choose k}_q</math>,
which gives another interpretation of the identity
:<math>{m \choose r}_q = {m-1 \choose r}_q + q^{m-r}{m-1 \choose r-1}_q.</math>
as counting the ''(r-1)''-dimensional subspaces of ''(m-1)''-dimensional projective space by fixing a hyperplane, counting such subspaces contained in that hyperplane, and then counting the subspaces not contained in the hyperplanes which are in bijective correspondence with the ''(r-1)''-dimensional affine subspaces of the space obtained by treating this fixed hyperplane as the hyperplane at infinity.
In the conventions common in applications to [[quantum groups]], a slightly different definition is used; the quantum binomial coefficient there is
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