2. Let $[x]$ denote the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. Then
$$
\begin{array}{l}
{\left[\frac{2009 \times 83}{2009}\right]+\left[\frac{2010 \times 83}{2009}\right]+\cdots+\left[\frac{4017 \times 83}{2009}\right]} \\
=(\quad) .
\end{array}
$$
(A) 249075
(B) 250958
(C) 174696
(D) 251000
2.A.

Original expression
$$
\begin{aligned}
= & {\left[\frac{(2009+0) \times 83}{2009}\right]+\left[\frac{(2009+1) \times 83}{2009}\right]+} \\
& \cdots+\left[\frac{(2009+2008) \times 83}{2009}\right] \\
= & 83+\left[\frac{0 \times 83}{2009}\right]+83+\left[\frac{1 \times 83}{2009}\right]+\cdots+ \\
& 83+\left[\frac{2008 \times 83}{2009}\right] \\
= & 83 \times 2009+\frac{0 \times 83}{2009}+\frac{1 \times 83}{2009}+\cdots+ \\
& \frac{2008 \times 83}{2009}-\left\{\frac{0 \times 83}{2009}\right\}-\left\{\frac{1 \times 83}{2009}\right\}-\cdots- \\
& \left\{\frac{2008 \times 83}{2009}\right\} \\
= & 2009 \times 83+\frac{83}{2009}(1+2+\cdots+2008)- \\
& \left\{\frac{0 \times 83}{2009}\right\}-\left\{\frac{1 \times 83}{2009}\right\}-\cdots-\left\{\frac{2008 \times 83}{2009}\right\} \\
= & 2009 \times 83+83 \times 1004-\left\{\frac{0 \times 83}{2009}\right\}- \\
& \left\{\frac{1 \times 83}{2009}\right\}-\cdots-\left\{\frac{2008 \times 83}{2009}\right\} .
\end{aligned}
$$

Clearly, 2009 and 83 are coprime, and $0 \times 83, 1 \times 83, \cdots, 2008 \times 83$ when divided by 2009 yield 2009 different remainders. Therefore,
$$
\begin{array}{l}
\quad\left\{\frac{0 \times 83}{2009}\right\}+\left\{\frac{1 \times 83}{2009}\right\}+\cdots+\left\{\frac{2008 \times 83}{2009}\right\} \\
\quad=\frac{0+1+\cdots+2008}{2009} \\
=1004 .
\end{array}
$$

Thus, the original expression $=2009 \times 83+83 \times 1004-1004$
$$
\begin{array}{l}
=166747+82328 \\
=249075 .
\end{array}
$$