The Basic Rules of Counting

# n n 1 n 2 3 2 1 1 1 2 2 3 6

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Unformatted text preview: − 1) × (n − 2) × . . . × 3 × 2 × 1 Example: Finishing orders in race with 8 runners, no ties allowed? 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40, 320 Notation: “n factorial” n! = n × (n − 1) × (n − 2) × . . . × 3 × 2 × 1 1! = 1, 2! = 2, 3! = 6, 4! = 24, . . ., Math 30530 (Fall 2012) Counting September 13, 2013 4 / 12 Arranging objects in a row In how many ways can n different, distinct objects be lined up in a row? n options for ﬁrst item, then n − 1 for second (regardless of what was chosen ﬁrst), then n − 2 for second, etc. So by product rule, ﬁnal count is n × (n − 1) × (n − 2) × . . . × 3 × 2 × 1 Example: Finishing orders in race with 8 runners, no ties allowed? 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40, 320 Notation: “n factorial” n! = n × (n − 1) × (n − 2) × . . . × 3 × 2 × 1 1! = 1, 2! = 2, 3! = 6, 4! = 24, . . ., 34! ≈ 29, 500, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000 Math 30530 (Fall 2012) Counting September 13, 2013 4 / 12 Arranging objects in a row In how many ways can n different, distinct objects be lined up in a row? n options for ﬁrst item, then n − 1 for second (regardless of what was chosen ﬁrst), then n − 2 for second, etc. So by product rule, ﬁnal count is n × (n − 1) × (n − 2) × . . . × 3 × 2 × 1 Example: Finishing orders in race with 8 runners, no ties allowed? 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40, 320 Notation: “n factorial” n! = n × (n − 1) × (n − 2) × . . . × 3 × 2 × 1 1! = 1, 2! = 2, 3! = 6, 4! = 24, . . ., 34! ≈ 29, 500, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000 Convention: 0! = 1 Math 30530 (Fall 2012) Counting September 13, 2013 4 / 12 Selecting k things from n, WITHOUT REPLACEMENT In how many ways can we pull out k distinct items from among n different, distinct objects? Math 30530 (Fall 2012) Counting September 13, 2013 5 / 12 Selecting k things from n, WITHOUT REPLACEMENT In how many ways can we pull out k distinct items from among n different, distinct objects? Order matters: (the k items have to be lined up in a row) n(n − 1) . . . (n − (k − 1)) = Math 30530 (Fall 2012) Counting n! (sometimes n Pk ) (n...
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