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boxeskey - Number of drawers = 20(corresponding to the...

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McCombs Math 381 Boxes 1 Distinguishable objects into distinguishable boxes. The number of ways to place n distinguishable objects into k distinguishable boxes, such that n 1 objects go in the first box, n 2 in the second, . .. and n k go in the k th box is given by n ! n 1 ! n 2 ! n 3 ! ! ! ! n k ! . Basic Idea: How many anagrams can be formed from GOOGLE ? Indistinguishable Objects into Distinguishable Boxes The number of different ways to distribute n indistinguishable balls into k distinguishable boxes is C n + k ! 1, k ! 1 ( ) = C n + k ! 1, n ( ) . Basic Idea: How many ways can you arrange n ! 1 repeated letters and k repeated letters? Example: Suppose a store sells 20 varieties of donuts. How many ways can you choose 12 donuts according to the given constraint? (i) no 2 donuts of the same variety P 20,12 ( ) = 20! 8! (ii) all donuts of the same variety P 20,1 ( ) = 20 (iii) no restrictions

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Unformatted text preview: Number of drawers = 20 (corresponding to the donut types) Number of dividers = 19 Number of donuts chosen = 12 Total number of solutions = C 12 + 20 ! 1,12 ( ) = 31 12 " # \$ % & ’ = 31 19 " # \$ % & ’ = 141120525 (iv) at least two varieties = total possible – exactly 1 variety 31 12 ! " # \$ % & ’ 20 = 141120505 McCombs Math 381 Boxes 2 (v) at least 6 glazed donuts Number of drawers = 20 (corresponding to the donut types) Number of dividers = 19 Number of donuts chosen = 12 – 6 = 6 Total number of solutions = C 6 + 20 ! 1,6 ( ) = 25 6 " # \$ % & ’ = 25 19 " # \$ % & ’ = 177100 (vi) no more than 6 glazed donuts = total – at least 6 glazed donuts – exactly 6 glazed donuts = 31 12 ! " # \$ % & ’ 25 6 ! " # \$ % & + 24 6 ! " # \$ % & = 141078021...
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boxeskey - Number of drawers = 20(corresponding to the...

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