17_Counting - The number of permutations of n objects,...

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Generalized Permutations and Combinations 10/20/11 Counting with n distinct objects
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Combinations with repetition Example : Three schools -- A, B and C -- are competing for a grand prize in a science fair competition. There are two judges. Each judge, anonymously, recommends one of the two schools. Calculate the number of ways the judges recommend the schools. The same as the number of 4-bit strings with exactly two 1s. # bit positions = # schools -1 + # of judges Theorem : C(n-1+r,r) is the number of r-combinations of n elements with repetition.
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Section 6.5, Problem 9 A bagel shop has eight different types of bagels. How many ways are there to choose
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Egg Salty Free choice from 7 types (other than salty)
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Permutations of indistinguishable objects Example : # of strings obtained by permuting the letters of the word "DISCR" is 5! Alternative calculation:
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Example 2 : calculate the # of strings obtained by permuting the letters of the word "DISCRETE". Theorem 3, Section 5.5
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Unformatted text preview: The number of permutations of n objects, where there are n 1 indistinguishable objects of type 1, n 2 indistinguishable objects of type 2, . .., and n k indistinguishable objects of type k such that n 1 +n 2 +...+n k = n, Distributing objects into boxes Many counting problems can be solved by enumerating the ways objects can be placed into boxes, where the order of placing objects within a box does not matter. There are four possibilities. we will look at placing distinguishable objects into distinguishable boxes (DODB) and indistinguishable objects into distinguishable boxes (IODB). Theorem 4. The number of ways to distribute n distinguishable objects into k distinct boxes so that n i objects are placed in box i, i=1, . .., k, and n 1 +...+n k = n, is Distinguishable objects into distinguishable boxes (DODB) Example : count the number of 5-card poker hands for 4 players in a game. Assume that a standard deck of cards is used....
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17_Counting - The number of permutations of n objects,...

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