= Dimension of a bounded-total-degree polynomial space
{title2=$\dim\mathcal P_{\leq d}(k^n)=\binom{n+d}{n}$}
The <vector space> of formal <polynomials> over a <field> $k$ in $n$ variables with <total degree of a polynomial> at most $d$ has the <monomial> basis $x_1^{a_1}\cdots x_n^{a_n}$ for nonnegative exponents with $\sum_i a_i\leq d$. Introducing a slack exponent and using <stars and bars> counts $\binom{n+d}{n}$ basis elements. These are formal <polynomials>; over a <finite field>, distinct formal <polynomials> need not define distinct <polynomial functions> unless suitable degree restrictions hold.
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