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Unformatted text preview: Power 12 / 22 Case 1: Exactly eight votes precede France How can we get 8 votes? If France is preceded by:
Case 1a: Germany, Belgium, and the Netherlands (4, 2, 2)
Case 1b: Italy, Belgium, and the Netherlands (4,2,2)
Case 1c: Germany and Italy (4, 4) For each of the three above subcases we can use counting techniques
to …gure out the number of ways this can happen. Van Essen (U of A) Power 13 / 22 Case 1a: Germany, Belgium, and the Netherlands precede
France To count the total number of orderings that are included in each
subcase we need to count: (1) how many ways can we arrange the
countries that precede France; (2) how many ways can we arrange the
countries that come after France; and (3) use the multiplication
principle to get the total.
France is preceded by Germany, Belgium, and the Netherlands
The number of distinct ways we can order these three countries is 3!
The number of ways we can arrange the countries that come after
France (Luxembourg and Italy) is 2!
Therefore the total number of pivotal orderings where France is
preceded by Germany, Belgium, and the Netherlands is 3!2! = 12 Van Essen (U of A) Power 14 / 22 Case 1b: Italy, Belgium, and the Netherlands precede
France France is preceded by Italy, Belgium, and the Netherlands
The number of distinct ways we can order these three countries is 3!
The number of ways we can arrange the countries that come after
France is 2!
Therefore the total number of pivotal orderings for case 1b 3!2! = 12 Van Essen (U of A) Power 15 / 22 Case 1c: Italy and Germany precede France France is preceded by Italy and Germany
The number of distinct ways we can order these two countries is 2!
The number of ways we can arrange the countries that come after
France is 3!
Therefore the total number of pivotal orderings for case 1b 2!3! = 12 Van Essen (U of A) Power 16 / 22 Case 1 Case 1a: 12 orderings
Case 1b: 12 orderings
Case 1c: 12 orderings
Therefore the total number of pivotal orderings for case 1 (8 votes) is
36 Van Essen (U of A) Power 17 / 22 Case 2: Exactly nine votes precede France How can we get 9 votes? If France is preceded by:
Case 2a: Germany, Be...
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 Spring '12
 Vanessen
 The Land

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