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Chapter 10: Problem 62
Give an example of two events that are not mutually exclusive.
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Convert the equation $$ 4 x^{2}+y^{2}-24 x+6 y+9=0 $$ to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of the foci. (Section 9.1, Example 5).
Use mathematical induction to prove that each statement is true for every positive integer \(n\). $$n+2 > n$$
Use this information to solve Exercises \(47-48 .\) The mathematics department of a college has 8 male professors, 11 female professors, 14 male teaching assistants, and 7 female teaching assistants. If a person is selected at random from the group, find the probability that the selected person is a professor or a female.
Will help you prepare for the material covered in the next section. Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\). $$\begin{array}{l} (a+b)^{1}=a+b \\ (a+b)^{2}=a^{2}+2 a b+b^{2} \\ (a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\ (a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\ (a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5} \end{array}$$ Describe the pattern for the exponents on \(a\).
Use mathematical induction to prove that each statement is true for every positive integer \(n\). 2 is a factor of \(n^{2}-n\)
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