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COT3100Hmk03Sol - COT 3100 Fall 2002 Homework#3 Solutions 1...

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COT 3100 Fall 2002 Homework #3 Solutions 1) Each of the following questions deals with permutations of TENNESSEE. a) How many permutations are there of the word TENNESSEE? Using the permutation formula derived in class, the answer is 9!/(4!2!2!) = 3780 b) How many of the permutations of TENNESSEE do not contain any occurences of contiguous vowels? Consider one ordering of the consonants: T , N , N , S , S Now, consider inserting vowels into this ordering. You can ONLY insert one vowel per gap. The gaps are illustrated below: __ T __ N __ N __ S __ S __ Out of these gaps, you must choose 4 of them to insert Es. This can be done in 6 C 4 ways. Now, since each vowel is an E, each of these ways corresponds to exactly one permutation of letters without adjacent Es. But, we only considered the number of permutations for one ordering of the consonants. There are actually 5!/(2!2!) possible ordering of the consonants using the permutation formula. For each of these, we have 6 C 4 arrangements of vowels. Thus, the final answer is the product of these: ( 6 C 4 )(5!/(2!2!)) = 450 c) How many permutations of TENNESSEE contain the substring EEEE? Treat the four letters EEEE as one single superletter. Then the letters we are permuting are T, N, N, S, S, EEEE. Using the permutation formula, there are 6!/(2!2!) = 180 of these permutations d) How many permutations of TENNESSEE contain all of the consonants in a single block? Now, treat the consonants TNNSS as a superletter. So now we are permuting the letters E , E , E , E , TNNSS. There are 5!/4! = 5 ways to do this. BUT, for each of these ways, there are 5!/(2!2!) valid permutations of the superletter. Thus, the final answer is 5(5!/(2!2!)) = 150 permutations.
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2) The local ice cream shop serves 31 different flavors of ice cream. A sundae includes one scoop of any flavor ice cream, along with a syrup and a topping, both of which are optional. There are 5 different types of syrups and 12 different types of toppings.
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