Mock AIME 2 Pre 2005 Problems/Problem 7

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Solution

The number of ways to choose 2 suits: (42)=6

The number of ways to distribute 26 cards to Reid and Po-Ru: (2613)

The number of ways to distribute the remaining 26 cards to Anders and Aaron: (2613)

Total number of ways: \[N = 6 \times \binom{26}{13} \times \binom{26}{13}\] Given (2613)mod1000=576: \[N = 6 \times 576 \times 576 \equiv 6 \times 331776 \equiv 6 \times 776 \equiv 4560 \equiv 560 \pmod{1000}\] \[\boxed{560}\]