The Basic Rules of Counting

last nk example 8 digit numbers with prime digits 48

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Unformatted text preview: 8 digit numbers with prime digits? 48 Example: In a group of 23 people, how likely is it that they all have their birthdays on a different date? Math 30530 (Fall 2012) Counting September 13, 2013 9 / 12 Selecting k items from n, WITH REPLACEMENT In how many ways can we pull out k items from among n different, distinct objects, if each time we pull out an item, we note it and put it back? Order matters: (we produce a list (first item, second, . . ., last)) nk Example: 8 digit numbers with prime digits? 48 Example: In a group of 23 people, how likely is it that they all have their birthdays on a different date? ways of choosing 23 different dates, in order ways of choosing 23 dates, in order 365×364×...×343 = 365×365×...×365 ≈ .4927 < 50% Math 30530 (Fall 2012) Counting September 13, 2013 9 / 12 Selecting k items from n, WITH 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 10 / 12 Selecting k items from n, WITH REPLACEMENT In how many ways can we pull out k distinct items from among n different, distinct objects? Order doesn’t matter: (we just note how many of the first item, how many of the second, etc.) n+k −1 k Math 30530 (Fall 2012) Counting September 13, 2013 10 / 12 Selecting k items from n, WITH REPLACEMENT In how many ways can we pull out k distinct items from among n different, distinct objects? Order doesn’t matter: (we just note how many of the first item, how many of the second, etc.) n+k −1 k Example: Select 8 single-digit primes, no particular order? 11 8 Math 30530 (Fall 2012) = 165 Counting September 13, 2013 10 / 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? 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 Math 30530 (Fall 2012) = Counting 88 36 ≈ 6 × 1034 September 13, 2013 11 / 12 Some examples I hav...
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