Chapter 6: Q7E (page 413)
Find the number of 5-permutations of a set with nine elements.
The resultant answer is 15,20..
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How many permutations of the letters \(ABCDEFGH\) contain
a) the string \(ED\)?
b) the string \(CDE\)?
c) the strings \(BA\) and \(FGH\)?
d) the strings \(AB\;,\;DE\) and \(GH\)?
e) the strings \(CAB\) and \(BED\)?
f) the strings \(BCA\) and \(ABF\)?
How many permutations of the letters \(ABCDEFG\) contain
a) the string \(BCD\)?
b) the string \(CFGA\)?
c) the strings \(BA\) and \(GF\)?
d) the strings \(ABC\)and \(DE\)?
e) the strings \(ABC\)and \(CDE\)?
f) the strings \(CBA\)and \(BED\)?.
Explain how to prove Pascal’s identity using a combinatorial argument.
A test containstrue/false questions. How many different ways can a student answer the questions on the test, if answers may be left blank?
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.
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