Chapter 6: Q21E (page 432)
How many ways are there to distribute six indistinguishable balls into nine distinguishable bins?
There are 3003 ways to distribute six indistinguishable balls into nine distinguishable bins
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8. How many different ways are there to choose a dozen donuts from the 21 varieties at a donut shop?
7. How many ways are there to select three unordered elements from a set with five elements when repetition is allowed?
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.
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) \({\rm{9}}\)
Explain how to prove Pascal’s identity using a combinatorial argument.
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