Chapter 6: Q45E (page 397)
How many sections of size \(34\) are needed to accommodate all the students?
\(40\) sections are needed to accommodate all the students.
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a) State the generalized pigeonhole principle.
b) Explain how the generalized pigeonhole principle can be used to show that among any 91 integers, there are at least ten that end with the same digit.
How can you find the number of bit strings of length ten that either begin with 101 or end with 010 ?
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.
How many bit strings of length \({\rm{10}}\) over the alphabet \({\rm{\{ a,b,c\} }}\) have either exactly three \({\rm{a}}\)s or exactly four \({\rm{b}}\)s?
How many different permutations are there of the set {a,b,c,d,e,f,g} ?
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