= Solution
Let $S=k[x_0,\ldots,x_n]$ and let $B=S/I(X)$ be the <homogeneous coordinate ring> of a nonempty <projective variety>. Its <Hilbert function> is $h_X(t)=\dim_k B_t$. For all sufficiently large integers $t$, this function agrees with a <polynomial> $P_X(t)$, the <Hilbert polynomial>.
The <dimension of an algebraic set>[dimension] $r$ is the largest length of a strict chain of nonempty <irreducible topological space>[irreducible] closed <subvarieties> of $X$. Equivalently $r=\dim B-1=\deg P_X$. The degree of the embedded <variety> is the integer specified by the leading term
$$
\boxed{P_X(t)=\frac{\deg X}{r!}t^r+\text{lower-degree terms}.}
$$
Geometrically it is the number of <intersection> <points> with a general <linear subspace> of codimension $r$, counted with <intersection multiplicities>. The <polynomial>'s degree is the <dimension of an algebraic set>[dimension] $r$; the <variety>'s degree is the <leading coefficient> multiplied by $r!$. Keeping those two meanings separate is important in the next computation.
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