What is the remainder when $301^4$ is divided by 10,000?
By the Binomial Theorem, we have  \begin{align*}
301^4 &= (3(100) + 1)^4\\
&= \binom40 \cdot 3^4 \cdot 100^4 \cdot 1^0 + \binom41 \cdot 3^3 \cdot 100^3 \cdot 1^1 \\
&\qquad\qquad+ \binom42 \cdot 3^2 \cdot 100^2 \cdot 1^2+ \binom43 \cdot 3^1 \cdot 100^1 \cdot 1^3 \\
&\qquad\qquad+ \binom44 \cdot 3^0 \cdot 100^0 \cdot 1^4.
\end{align*} All but the last two terms are divisible by $10000=100^2$, so we only have to consider the remainder when the last two terms are divided by 10,000. The last two terms are  \begin{align*}
\binom43 &\cdot 3^1 \cdot 100^1 \cdot 1^3 + \binom44 \cdot 3^0 \cdot 100^0 \cdot 1^4\\
&= 4 \cdot 3 \cdot 100 \cdot 1 + 1 \cdot 1 \cdot 1 \cdot 1\\
& = 1200 + 1 = \boxed{1201}.\end{align*}