Difference between revisions of "Mock AIME 2 Pre 2005 Problems/Problem 7"
Anyu tsuruko (talk | contribs) (Created page with "== Solution == The number of ways to choose 2 suits: \(\binom{4}{2} = 6\) The number of ways to distribute 26 cards to Reid and Po-Ru: \(\binom{26}{13}\) The number of...") |
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Latest revision as of 16:34, 24 May 2024
Solution
The number of ways to choose 2 suits: \(\binom{4}{2} = 6\)
The number of ways to distribute 26 cards to Reid and Po-Ru: \(\binom{26}{13}\)
The number of ways to distribute the remaining 26 cards to Anders and Aaron: \(\binom{26}{13}\)
Total number of ways:
![\[N = 6 \times \binom{26}{13} \times \binom{26}{13}\]](http://latex.artofproblemsolving.com/6/1/9/61995d940ccd2368cd78cc38140462c36fe526dc.png)
Given \(\binom{26}{13} \mod 1000 = 576\):
![\[N = 6 \times 576 \times 576 \equiv 6 \times 331776 \equiv 6 \times 776 \equiv 4560 \equiv 560 \pmod{1000}\]](http://latex.artofproblemsolving.com/9/5/4/954aa06cb9aa089786dfff7a5e1f5b2f7b634f99.png)
![\[\boxed{560}\]](http://latex.artofproblemsolving.com/2/c/2/2c2907515caa2d67fbcb429d1960f07e45bfb1ef.png)
