Chapter 6: Q43E (page 415)
How many ways are there for a horse race with three horses to finish if ties are possible? (Note: Two or three horses may tie.)
The required number of ways are\(13\).
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Let. S = {1,2,3,4,5}
a) List all the 3-permutations of S.
b) List all the 3 -combinations of S.
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?
When the numbers from \({\rm{1}}\) to \({\rm{1000}}\) are written out in decimal notation, how many of each of these digits are used?
a) \({\rm{0}}\)
b) \({\rm{1}}\)
c) \({\rm{2}}\)
d) \({\rm{9}}\)
In this exercise we will count the number of paths in the\(xy\)plane between the origin \((0,0)\) and point\((m,n)\), where\(m\)and\(n\)are nonnegative integers, such that each path is made up of a series of steps, where each step is a move one unit to the right or a move one unit upward. (No moves to the left or downward are allowed.) Two such paths from\((0,0)\)to\((5,3)\)are illustrated here.
a) Show that each path of the type described can be represented by a bit string consisting of\(m\,\,0\)s and\(n\,\,1\)s, where a\(0\)represents a move one unit to the right and a\(1\)represents a move one unit upward.
b) Conclude from part (a) that there are \(\left( {\begin{array}{*{20}{c}}{m + n}\\n\end{array}} \right)\) paths of the desired type.
a) How many ways are there to deal hands of five cards to six players from a standard 52-card deck?
b) How many ways are there to distribute n distinguishable objects into kdistinguishable boxes so thatobjects are placed in box i?
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