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Product rule 2: if an experiment is performed in m stages, with
n1 outcomes for the ﬁrst stage, and
n2 outcomes for the second, REGARDLESS OF FIRST, . . ., and
nm outcomes for the mth, REGARDLESS OF ALL PREVIOUS, then the total number of outcomes for the experiment is
n1 n2 . . . nm
Math 30530 (Fall 2012) Counting September 13, 2013 3 / 12 Arranging objects in a row
In how many ways can n different, distinct objects be lined up in a row? 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 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 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 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...
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This note was uploaded on 01/31/2014 for the course MATH 30530 taught by Professor Hind,r during the Fall '08 term at Notre Dame.
 Fall '08
 Hind,R
 Counting, Probability

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