Unformatted text preview: e 36 identical prizes to distribute to the class (53 people). All I
care about is how many prizes each student gets. How many
possible ways to distribute are there?
Here n = 53 (I’m choosing from pool of students), r = 36 (I’m
choosing students to give prizes to), and I’m choosing with
replacement, order not mattering, so solution is
53 + 36 − 1
36 = 88
36 ≈ 6 × 1034 What’s the probability that Zeke doesn’t get a prize, assuming all
ways of distribution equally likely? Math 30530 (Fall 2012) Counting September 13, 2013 11 / 12 Some examples
I have 36 identical prizes to distribute to the class (53 people). All I
care about is how many prizes each student gets. How many
possible ways to distribute are there?
Here n = 53 (I’m choosing from pool of students), r = 36 (I’m
choosing students to give prizes to), and I’m choosing with
replacement, order not mattering, so solution is
53 + 36 − 1
36 = 88
36 ≈ 6 × 1034 What’s the probability that Zeke doesn’t get a prize, assuming all
ways of distribution equally likely?
87
88
/
36
36 Math 30530 (Fall 2012) Counting ≈ .59 September 13, 2013 11 / 12 Summary of counting problems
Sum rule: A OR B ? Add
Product rule: A THEN B ? Multiply
Overcount rule: Each item counted too many times? Divide
Arranging n items in order: n!
Selecting k items from n, WITHOUT REPLACEMENT
ORDER MATTERS: n!
(n−k )! ORDER DOESN’T MATTER: n
k = n!
k !(n−k )! Selecting k items from n, WITH REPLACEMENT
ORDER MATTERS: nk
ORDER DOESN’T MATTER: n +k −1
k Partitioning n into classes of size n1 , n2 , . . ., nr , OR arranging n items
in a row when there are n1 of ﬁrst type, n2 of second, etc., and we can’t
n
tell the difference within types: n1 ,n2n nr = n1 !n2 !!...nr !
,...,
Math 30530 (Fall 2012) Counting September 13, 2013 12 / 12...
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 Fall '08
 Hind,R
 Counting, Probability

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