Geodesic
Asymptotic behavior of the number of certain $k$-dimensional subspaces over a~finite field
Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 729-736.

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For some values of $k$, we find the asymptotic behavior, as $n\to\infty$, of the probability that a subspace, whose choice is random and equiprobable, chosen among the set of all different $k$-dimensional subspaces of an $n$-dimensional vector space over a finite field, has a given weight $\omega\in\{1,2,\dots,n\}$. In particular, for $\omega\in\{1,2\}$, this probability can have exponential behavior. 
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     author = {V. I. Masol},
     title = {Asymptotic behavior of the number of certain $k$-dimensional subspaces over a~finite field},
     journal = {Matemati\v{c}eskie zametki},
     pages = {729--736},
     publisher = {mathdoc},
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     number = {5},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a8/}
}
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V. I. Masol. Asymptotic behavior of the number of certain $k$-dimensional subspaces over a~finite field. Matematičeskie zametki, Tome 59 (1996) no. 5, pp. 729-736. http://geodesic.mathdoc.fr/item/MZM_1996_59_5_a8/

[1] Endryus G., Teoriya razbienii, Nauka, M., 1982