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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 Fall '08
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 Counting, Probability

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