Chapter 6: Q1E (page 413)
List all the permutations of {a,b,c}.
The resultant answer is abc, acb, bac, bca, cab, cba.
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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\(6.3\).
A test containstrue/false questions. How many different ways can a student answer the questions on the test, if answers may be left blank?
a) What is Pascal’s triangle?
b) How can a row of Pascal’s triangle be produced from the one above it?
The internal telephone numbers in the phone system on a campus consist of five digits, with the first digit not equal to zero. How many different numbers can be assigned in this system?
What is the coefficient of?
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