Chapter 6: Q4E (page 413)
Let. S = {1,2,3,4,5}
a) List all the 3-permutations of S.
b) List all the 3 -combinations of S.
(a) The resultant answer is:
(b) The resultant answer is:
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Find the value of each of these quantities:
a) P (6,3)
b) P (6,5)
c) P (8,1))
d) P 8,5)
e) P (8,8)
f) P (10,9)
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.
Show that for all positive integers nand all integers kwith .
Let\(n\)be a positive integer. Show that\(\left( {\begin{array}{*{20}{c}}{2n}\\{n + 1}\end{array}} \right) + \left( {\begin{array}{*{20}{c}}{2n}\\n\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{2n + 2}\\{n + 1}\end{array}} \right)/2\).
A test containstrue/false questions. How many different ways can a student answer the questions on the test, if answers may be left blank?
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