Chapter 9

In Exercises 1 - 7, fill in the blanks. An ________ is an event whose result is uncertain, and the possible results of the event are called ________.

Fill in the blanks The ________ ________ ________ states that if there are \( m_1 \) ways for one event to occur and \( m_2 \) ways for a second event to occur, there are \( m_1 \cdot m_2 \) ways for both events to occur.

Fill in the blanks. The coefficients of a binomial expansion are called ________ ________.

Fill in the blanks The first step in proving a formula by ________ ________ is to show that the formula is true when \( n = 1 \).

Fill in the blanks. A sequence is called a ________ sequence if the ratios between consecutive terms are the same. This ratio is called the ________ ratio.

Fill in the blanks. An ________ ________ is a function whose domain is the set of positive integers.

Fill in the blanks. A sequence is called an ________ sequence if the differences between consecutive terms are the same. This difference is called the ________ difference.

In Exercises 1 - 7, fill in the blanks. The set of all possible outcomes of an experiment is called the ________ ________.

Fill in the blanks An ordering of \( n \) elements is called a ________ of the elements.

Fill in the blanks The ________ differences of a sequence are found by subtracting consecutive terms.

Fill in the blanks. The \( n \)th term of an arithmetic sequence has the form ________.

Fill in the blanks. The function values \( a_1, a_2, a_3, a_4, \cdots \) are called the ________ of a sequence.

In Exercises 1 - 7, fill in the blanks. To determine the ________ of an event, you can use the formula \( P\left(e\right) = \dfrac{n\left(E\right)}{n\left(S\right)} \), where \( n\left(E\right) \) is the number of outcomes in the event and \( n\left(S\right) \) is the number of outcomes in the sample space.

Fill in the blanks The number of permutations of \( n \) elements taken \( r \) at a time is given by the formula ________.

Fill in the blanks. The notation used to denote a binomial coefficient is ________ or ________.

Fill in the blanks A sequence is an ________ sequence if the first differences are all the same nonzero number.

Fill in the blanks. The formula for the sum of a finite geometric sequence is given by ________.

Fill in the blanks. If you know the \( n \)th term of an arithmetic sequence and you know the common difference of the sequence,you can find \( (n + 1)th \) term by using the ________ formula \( a_{n + 1} = a_n + d \).

Fill in the blanks. A sequence is a ________ sequence if the domain of the function consists only of the first \( n \) positive integers.

In Exercises 1 - 7, fill in the blanks. If \( P\left(E\right) = 0 \), then \( E \) is an ______ event, and if \( P\left(E\right) = 1 \), then \( E \) is a _______ event.

Fill in the blanks. When you write out the coefficients for a binomial that is raised to a power, you are ________ a ________.

Fill in the blanks If the ________ differences of a sequence are all the same nonzero number, then the sequence has a perfect quadratic model.

Fill in the blanks. The sum of the terms of an infinite geometric sequence is called a ________ ________.

Fill in the blanks. The formula \( S_n = \dfrac{n}{2}\left(a_1 + a_n\right) \) can be used to find the sum of the first \( n \) terms of an arithmetic sequence,called the ________ of a ________ ________ ________.

Fill in the blanks. If you are given one or more of the first few terms of a sequence, and all other terms of the sequence are defined using previous terms, then the sequence is said to be defined ________.

In Exercises 1 - 7, fill in the blanks. If two events from the same sample space have no outcomes in common, then the two events are ________ ________.

Fill in the blanks When selecting subsets of a larger set in which order is not important, you are finding the number of ________ of \( n \) elements taken \( r \) at a time.

In Exercises 5 - 14, calculate the binomial coefficient. \( _5C_3 \)

In Exercises 5 - 10, find \( P_{k + 1} \) for the given \( P_k \). \( P_k = \dfrac{5}{k\left(k + 1\right)} \)

In Exercises 5 - 16, determine whether the sequence is geometric. If so, find the common ratio. \( 2, 10, 50, 250, \cdots \)

In Exercises 5 - 14, determine whether the sequence is arithmetic. If so, find the common difference. \( 10, 8, 6, 4, 2, \cdots \)

Fill in the blanks. If \( n \) is a positive integer, \( n \) ________ is defined as \( n ! = 1 \cdot 2 \cdot 3 \cdot 4 \cdots (n - 1) \cdot n \).

In Exercises 1 - 7, fill in the blanks. If the occurrence of one event has no effect on the occurrence of a second event, then the events are ________.

In Exercises 5 - 14, calculate the binomial coefficient. \( _8C_6 \)

In Exercises 5 - 10, find \( P_{k + 1} \) for the given \( P_k \). \( P_k = \dfrac{1}{2\left(k + 2\right)} \)

In Exercises 5 - 16, determine whether the sequence is geometric. If so, find the common ratio. \( 7, 21, 63, 189, \cdots \)

In Exercises 5 - 14, determine whether the sequence is arithmetic. If so, find the common difference. \( 4, 9, 14, 19, 24, \cdots \)

Fill in the blanks. The notation used to represent the sum of the terms of a finite sequence is ________ ________ or sigma notation.

In Exercises 1 - 7, fill in the blanks. The ________ of an event \( A \) is the collection of all outcomes in the sample space that are not in \( A \).

In Exercises 5 - 14, calculate the binomial coefficient. \( _{12}C_0 \)

In Exercises 5 - 14, determine whether the sequence is arithmetic. If so, find the common difference. \( 4, 9, 14, 19, 24, \cdots \)

Fill in the blanks. For the sum \( \displaystyle \sum_{i=1}^{n} {a_i} \), \( i \) is called the ________ of summation, \( n \) is the ________ limit of summation, and 1 is the ________ limit of summation.

Match the probability formula with the correct probability name. (a) Probability of the union of two events \( \quad (i) P(A) + P(B) \) (b) Probability of mutually exclusive \( \quad \quad (ii) P(A') = 1 - P(A) \) (c) Probability of independent events \( \quad \quad (iii) P(A. \cup B) = P(A) + P(B) - P(A \cap B) \) (d) Probability of a complement \( \quad \quad \quad(iv) P(A \) and \( B) = P(A) \cdot P(B) \)

In Exercises 5 - 14, calculate the binomial coefficient. \( _{20}C_{20} \)

In Exercises 5 - 10, find \( P_{k + 1} \) for the given \( P_k \). \( P_k = \dfrac{k}{3} \left(2k + 1\right) \)

In Exercises 5 - 16, determine whether the sequence is geometric. If so, find the common ratio. \( 25, 20, 15, 10, \cdots \)

Fill in the blanks. The sum of the terms of a finite or infinite sequence is called a ________.

In Exercises 9 - 14, determine the sample space for the experiment. A coin and a six-sided die are tossed.

In Exercises 5 - 14, calculate the binomial coefficient. \( _{20C_{15} \)

In Exercises 5 - 10, find \( P_{k + 1} \) for the given \( P_k \). \( P_k = \dfrac{3}{\left(k + 2\right)\left(k + 3\right)} \)