Chapter 6: Q15E (page 413)
In how many ways can a set of five letters be selected from the English alphabet?
There are 65,780 ways can a set of five letters be selected from the English alphabets.
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What is the coefficient ofin the expansion of?
Find the expansion of.
A group contains n men and n women. How many ways are there to arrange these people in a row if the men and women alternate?
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.
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?
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