The Basic Rules of Counting

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online