MathDB

4

Part of 2006 AIME Problems

Problems(2)

Factorial Frenzy

Source: 2006 AIME A Problem 4

3/9/2006
Let NN be the number of consecutive 0's at the right end of the decimal representation of the product 1!×2!×3!×4!⋯99!×100!.1!\times2!\times3!\times4!\cdots99!\times100!. Find the remainder when NN is divided by 1000.
factorialnumber theoryprime factorizationAMC
Permutations of numbers

Source: 2006 AIME II 4

3/28/2006
Let (a1,a2,a3,...,a12)(a_1,a_2,a_3,...,a_{12}) be a permutation of (1,2,3,...,12)(1,2,3,...,12) for which a1>a2>a3>a4>a5>a6 and a6<a7<a8<a9<a10<a11<a12. a_1>a_2>a_3>a_4>a_5>a_6 \text{ and } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}. An example of such a permutation is (6,5,4,3,2,1,7,8,9,10,11,12)(6,5,4,3,2,1,7,8,9,10,11,12). Find the number of such permutations.
AMC