Chapter 6: Q24E (page 396)
How many positive integers between \(1000\) and \(9999\) inclusive are divisible by \(5\)and by \(7\) ?
\(257\) integers are divisible by \(5\) and \(7\).
Over 22 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
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 \(BED\)?.
a) State the pigeonhole principle.
b) Explain how the pigeonhole principle can be used to show that among any 11 integers, at least two must have the same last digit.
Prove that if\(n\)and\(k\)are integers with\(1 \le k \le n\), then\(k \cdot \left( {\begin{array}{*{20}{l}}n\\k\end{array}} \right) = n \cdot \left( {\begin{array}{*{20}{l}}{n - 1}\\{k - 1}\end{array}} \right)\),
a) using a combinatorial proof. [Hint: Show that the two sides of the identity count the number of ways to select a subset with\(k\)elements from a set with n elements and then an element of this subset.]
b) using an algebraic proof based on the formula for\(\left( {\begin{array}{*{20}{l}}n\\r\end{array}} \right)\)given in Theorem\(2\)in Section\(6.3\).
In how many ways can a set of five letters be selected from the English alphabet?
What is the coefficient ofin the expansion of
What do you think about this solution?
We value your feedback to improve our textbook solutions.
