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  "body": "\n## Definition\n\nLet $q$ be an indeterminate and let $n, k$ be integers with $n \\ge 0$.\nFor $0 \\le k \\le n$, the *Gaussian binomial coefficient* is\n$$\\binom{n}{k}_q = \\frac{(1 - q^n)(1 - q^{n-1}) \\cdots (1 - q^{n-k+1})}{(1 - q)(1 - q^2) \\cdots (1 - q^k)} = \\frac{[n]_q!}{[k]_q!\\, [n - k]_q!},$$\nwhere $[m]_q!$ is the q-factorial of the [[q-analog|GFSDMJHHVA]] node.\nFor $k \u003c 0$ or $k \u003e n$, $\\binom{n}{k}_q = 0$.\n\n## Remarks\n\nThe coefficients satisfy the q-Pascal rule $\\binom{n}{k}_q = \\binom{n-1}{k-1}_q + q^k \\binom{n-1}{k}_q$ for $n \\ge 1$.\nBy induction, $\\binom{n}{k}_q$ is a polynomial in $q$ with nonnegative integer coefficients, of degree $k(n - k)$.\nAt $q = 1$ it equals the [[binomial coefficient|0GBZM0MT0F]] $\\binom{n}{k}$.\nIf $q$ is a prime power, it is the number of $k$-dimensional subspaces of $\\F_q^n$.\n\nExample: $\\binom{4}{2}_q = \\frac{(1 - q^4)(1 - q^3)}{(1 - q)(1 - q^2)} = (1 + q^2)(1 + q + q^2) = 1 + q + 2q^2 + q^3 + q^4$.\nAt $q = 2$ this is $35$.\nCounting directly, $\\F_2^4$ has $\\frac{(2^4 - 1)(2^4 - 2)}{(2^2 - 1)(2^2 - 2)} = \\frac{15 \\cdot 14}{3 \\cdot 2} = 35$ subspaces of dimension $2$.\n\nNon-example: $q^{k(n-k)} \\binom{n}{k}$ also equals $\\binom{n}{k}$ at $q = 1$, but it is not the Gaussian binomial coefficient.\nFor $n = 4$, $k = 2$ and $q = 2$ it gives $96$, not $35$.\n\nThe coefficient of $q^j$ in $\\binom{n}{k}_q$ is the number of integer partitions of $j$ with at most $k$ parts, each at most $n - k$.\nAnother common notation is $\\left[ {n \\atop k} \\right]_q$.\n",
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