Chapter 6: Q1E (page 413)
List all the permutations of {a,b,c}.
The resultant answer is abc, acb, bac, bca, cab, cba.
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One hundred tickets, numbered \(1,2,3, \ldots ,100\), are sold to \(100\) different people for a drawing. Four different prizes are awarded, including a grand prize (a trip to Tahiti). How many ways are there to award the prizes if
a) there are no restrictions?
b) the person holding ticket \(47\) wins the grand prize?
c) the person holding ticket \(47\) wins one of the prizes?
d) the person holding ticket \(47\) does not win a prize?
e) the people holding tickets \(19\) and \(47\) both win prizes?
f) the people holding tickets \(19\;,\;47\)and \(73\) all win prizes?
g) the people holding tickets \(19\;,\;47\;,\;73\) and \(97\) all win prizes?
h) none of the people holding tickets \(19\;,\;47\;,\;73\) and \(97\) wins a prize?
i) the grand prize winner is a person holding ticket \(19\;,\;47\;,\;73\) or \(97\)?
j) the people holding tickets 19 and 47 win prizes, but the people holding tickets \(73\) and \(97\) do not win prizes?
Find the value of each of these quantities.
a) \(C(5,1)\)
b) \(C(5,3)\)
c) \(C(8,4)\)
d) \(C(8,8)\)
e) \(C(8,0)\)
f) \(C(12,6)\)
There are \({\rm{12}}\) signs of the zodiac. How many people are needed to guarantee that at least six of these people have the same sign?
a) Explain how to find a formula for the number of ways to select robjects from nobjects when repetition is allowed and order does not matter.
b) How many ways are there to select a dozen objects from among objects of five different types if objects of the same type are indistinguishable?
c) How many ways are there to select a dozen objects from these five different types if there must be at least three objects of the first type?
d) How many ways are there to select a dozen objects from these five different types if there cannot be more than four objects of the first type?
e) How many ways are there to select a dozen objects from these five different types if there must be at least two objects of the first type, but no more than three objects of the second type?
How many permutations of the letters \(ABCDEFG\) contain
a) the string \(BCD\)?
b) the string \(CFGA\)?
c) the strings \(BA\) and \(GF\)?
d) the strings \(ABC\)and \(DE\)?
e) the strings \(ABC\)and \(CDE\)?
f) the strings \(CBA\)and \(BED\)?.
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