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  "body": "\n## Summary\n\nThe Gaussian binomial coefficient $\\binom{n}{k}_q$ is the [[q-analog|GFSDMJHHVA]] of the [[binomial coefficient|0GBZM0MT0F]] $\\binom{n}{k}$, see the [[definition|TZ6Y9XXQJW]].\nIt is a polynomial in $q$ with nonnegative integer coefficients, and it equals $\\binom{n}{k}$ at $q = 1$.\nWhen $q$ is a prime power, it counts the $k$-dimensional subspaces of the [[vector space|000AYPA9AJ]] $\\F_q^n$.\nIts coefficients count the [[integer partitions|0JV30XN73Z]] whose [[Young diagram|096AHMB3P4]] fits in a $k \\times (n - k)$ box.\n\n## Explanations\n\n### L1 human\n\nThe ordinary binomial coefficient counts subsets of a set.\nThe Gaussian binomial coefficient counts subspaces of a vector space over a finite field with $q$ elements.\nFor example, $\\binom{4}{2}_q = 1 + q + 2q^2 + q^3 + q^4$.\nAt $q = 1$ it gives $6$, the number of $2$-element subsets of a $4$-element set.\nAt $q = 2$ it gives $35$, the number of planes through the origin in $\\F_2^4$.\n",
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