Difference between revisions of "2014 AIME II Problems/Problem 9"

Line 1: Line 1:
We know that a subset with less than 3 chairs cannot contain 3 adjacent chairs.  There are only 10 sets of 3 chairs so that they are all 3 adjacent.  There are 10 subsets of 4 chairs where all 4 are adjacent, and 50 where there are only 3 (10 * 5).  If there are 5 chairs, 10 have all 5 adjacent, 10 * 4 or 40 have 4 adjacent, and 10 * 5c2 or 100 have 3 adjacent.  With 6 chairs in the subset, 10 have all 6 adjacent, 10 * 3 or 30 have 5 adjacent, 10 * 4c2 or 60 have 4 adjacent, 10 * 3 / 2 or 15 have 2 groups of 3 adjacent chairs, and 10 * (5c2 - 3) or 70 have 1 group of 3 adjacent chairs.  All possible subsets with more than 6 chairs have at least 1 group of 3 adjacent chairs, so we add 10c7 or 120, 10c8 or 45, 10c9 or 10, and 10c10 or 1.  Adding, we get 10 + 10 + 50 + 10 + 40 + 100 + 10 + 30 + 60 + 15 + 70 + 120 + 45 + 10 + 1 = 581.
+
We know that a subset with less than 3 chairs cannot contain 3 adjacent chairs.  There are only 10 sets of 3 chairs so that they are all 3 adjacent.  There are 10 subsets of 4 chairs where all 4 are adjacent, and 10 * 5 or 50 where there are only 3.  If there are 5 chairs, 10 have all 5 adjacent, 10 * 4 or 40 have 4 adjacent, and 10 * 5c2 or 100 have 3 adjacent.  With 6 chairs in the subset, 10 have all 6 adjacent, 10 * 3 or 30 have 5 adjacent, 10 * 4c2 or 60 have 4 adjacent, 10 * 3 / 2 or 15 have 2 groups of 3 adjacent chairs, and 10 * (5c2 - 3) or 70 have 1 group of 3 adjacent chairs.  All possible subsets with more than 6 chairs have at least 1 group of 3 adjacent chairs, so we add 10c7 or 120, 10c8 or 45, 10c9 or 10, and 10c10 or 1.  Adding, we get 10 + 10 + 50 + 10 + 40 + 100 + 10 + 30 + 60 + 15 + 70 + 120 + 45 + 10 + 1 = 581.

Revision as of 20:18, 29 March 2014

We know that a subset with less than 3 chairs cannot contain 3 adjacent chairs. There are only 10 sets of 3 chairs so that they are all 3 adjacent. There are 10 subsets of 4 chairs where all 4 are adjacent, and 10 * 5 or 50 where there are only 3. If there are 5 chairs, 10 have all 5 adjacent, 10 * 4 or 40 have 4 adjacent, and 10 * 5c2 or 100 have 3 adjacent. With 6 chairs in the subset, 10 have all 6 adjacent, 10 * 3 or 30 have 5 adjacent, 10 * 4c2 or 60 have 4 adjacent, 10 * 3 / 2 or 15 have 2 groups of 3 adjacent chairs, and 10 * (5c2 - 3) or 70 have 1 group of 3 adjacent chairs. All possible subsets with more than 6 chairs have at least 1 group of 3 adjacent chairs, so we add 10c7 or 120, 10c8 or 45, 10c9 or 10, and 10c10 or 1. Adding, we get 10 + 10 + 50 + 10 + 40 + 100 + 10 + 30 + 60 + 15 + 70 + 120 + 45 + 10 + 1 = 581.

Invalid username
Login to AoPS