Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals:
$\textbf{(A)}\ (x-2)^4 \qquad \textbf{(B)}\ (x-1)^4 \qquad \textbf{(C)}\ x^4 \qquad \textbf{(D)}\ (x+1)^4 \qquad \textbf{(E)}\ x^4+1$
Let $y = x-1$.  Substitution results in
\[S = y^4 + 4y^3 + 6y^2 + 4y + 1\]
\[S = (y+1)^4\]
Substituting back results in
\[S = x^4\]
The answer is $\boxed{\textbf{(C)}}$.  This problem can also be solved by traditionally expanding and combining like terms (though it would take much longer).