Unformatted text preview: − k )! 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)) = n!
(sometimes n Pk )
(n − k )! Example: 1st, 2nd and 3th in race with 8 runners? 8 × 7 × 6 = 336 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)) = n!
(sometimes n Pk )
(n − k )! Example: 1st, 2nd and 3th in race with 8 runners? 8 × 7 × 6 = 336
Order doesn’t matter: (the k items are thrown together in a bag)
n(n − 1) . . . (n − (k − 1))
=
k! Math 30530 (Fall 2012) n
k Counting (“n choose k ”, sometimes n Ck ) 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)) = n!
(sometimes n Pk )
(n − k )! Example: 1st, 2nd and 3th in race with 8 runners? 8 × 7 × 6 = 336
Order doesn’t matter: (the k items are thrown together in a bag)
n(n − 1) . . . (n − (k − 1))
=
k! n
k (“n choose k ”, sometimes n Ck ) Example: Top three in race with eight runners? Math 30530 (Fall 2012) Counting 8
3 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)) = n!
(sometimes n Pk )
(n − k )! Example: 1st, 2nd and 3th in race with 8 runners? 8 × 7 × 6 = 336
Order doesn’t matter: (the k items are thrown together in a bag)
n(n − 1) . . . (n − (k − 1))
=
k! n
k (“n choose k ”, sometimes n Ck ) Example: Top three in race with eight runners? 8
3 Basic counting rule 3 — The overcount rule
If x is an initial count of some set of objects, and each object you
want to count ap...
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 Counting, Probability

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