Chapter 6: Q25E (page 430)
Show that and are logically equivalent
logically equivalent.
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How many subsets with an odd number of elements does a set withelements have?
Suppose that bis an integer with . Use the binomial theorem and the appropriate row of Pascal's triangle to find the base- bexpansion of [that is, the fourth power of the number in base-bnotation].
Prove the identity\(\left( {\begin{array}{*{20}{l}}n\\r\end{array}} \right)\left( {\begin{array}{*{20}{l}}r\\k\end{array}} \right) = \left( {\begin{array}{*{20}{l}}n\\k\end{array}} \right)\left( {\begin{array}{*{20}{l}}{n - k}\\{r - k}\end{array}} \right)\), whenever\(n\),\(r\), and\(k\)are nonnegative integers with\(r \le n\)and\(k{\rm{ }} \le {\rm{ }}r\),
a) using a combinatorial argument.
b) using an argument based on the formula for the number of \(r\)-combinations of a set with\(n\)elements.
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) Derive a formula for the number of permutations ofobjects of k different types, where there are indistinguishable objects of type one, indistinguishable objects of type two,..., and indistinguishable objects of type k.
b) How many ways are there to order the letters of the word INDISCREETNESS?
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