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a2-7-08

# a2-7-08 - Answers to Homework 7 February 2008 Section 15 1...

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Answers to Homework 7 February 2008 Section 15 1. Find all possible partitions of the set f 1 ; 2 ; 3 g . Classify them by how many sets are in the partition. One partition into one set ff 1 ; 2 ; 3 gg . Three partitions into two sets ff 1 g ; f 2 ; 3 gg , ff 2 g ; f 1 ; 3 gg , and ff 3 g ; f 1 ; 2 gg . One partition into three sets: ff 1 g ; f 2 g ; f 3 gg . Total of 1 + 3 + 1 = 5 partitions. Find all possible partitions of the set f 1 ; 2 ; 3 ; 4 g . One partition into one set: ff 1 ; 2 ; 3 ; 4 gg Four partitions with one one-element set: ff 1 g ; f 2 ; 3 ; 4 gg , ff 2 g ; f 1 ; 3 ; 4 gg , ff 3 g ; f 1 ; 2 ; 4 gg , ff 4 g ; f 1 ; 2 ; 3 gg Three partitions with two two-element sets: ff 1 ; 2 g ; f 3 ; 4 gg , ff 1 ; 3 g ; f 2 ; 4 gg , ff 1 ; 4 g ; f 2 ; 3 gg Six partitions into three sets: ff 1 ; 2 g ; f 3 g ; f 4 gg , ff 1 ; 3 g ; f 2 g ; f 4 gg , ff 1 ; 4 g ; f 2 g ; f 3 gg , ff 2 ; 3 g ; f 1 g ; f 4 gg , ff 2 ; 4 g ; f 1 g ; f 3 gg , ff 3 ; 4 g ; f 1 g ; f 2 gg One partition into four sets: ff 1 g ; f 2 g ; f 3 g ; f 4 gg Total of 1 + 4 + 3 + 6 + 1 = 15 partitions. 2. How many di/erent anagrams (including nonsensical words) can be made from each of the following : (a) STAPLE There are six distinct letters, so the °rst letter in the anagram can be any of 6 , the second letter any of the 5 remaining, and so on. The number of di/erent anagrams is 6! = 720 , which includes the original word STAPLE. (b) DISCRETE There are eight letters but two of them are the same. If you just look at rearrangements of the tiles, as it were, you will get the answer 8! .

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a2-7-08 - Answers to Homework 7 February 2008 Section 15 1...

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