AMC10 2011 B

Answer (D): In the expansion of $(2000+11)^{2011}$, all terms except $11^{2011}$ are divisible by $1000$, so the hundreds digit of $2011^{2011}$ is equal to that of $11^{2011}$. Furthermore, in the expansion of $(10+1)^{2011}$, all terms except $1^{2011}$, $\binom{2011}{1}(10)(1^{2010})$, and $\binom{2011}{2}(10)^2(1^{2009})$ are divisible by $1000$. Thus the hundreds digit of $11^{2011}$ is equal to that of $$ 1+\binom{2011}{1}(10)(1^{2010})+\binom{2011}{2}(10)^2(1^{2009}) $$ $$ =1+2011\cdot 10+2011\cdot 1005\cdot 100 $$ $$ =1+2011\cdot 100510. $$ Finally, the hundreds digit of this number is equal to that of $1+11\cdot 510=5611$, so the requested hundreds digit is $6$.