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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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View Full Document 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.
a)
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This document was uploaded on 07/14/2011.
 Spring '09

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