The Basic Rules of Counting

# Math 30530 fall 2012 counting september 13 2013 11

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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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