Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and... Problem 1.2P: How many outcome sequences are possible ten a die is rolled four times, where we say, for instance,... Problem 1.3P: Twenty workers are to be assigned to 20 different jobs, one to each job. How many different... Problem 1.4P: John, Jim, Jay, and Jack have formed a band consisting of 4 instruments if each of the boys can play... Problem 1.5P: For years, telephone area codes in the United States and Canada consisted of a sequence of three... Problem 1.6P: A well-known nursery rhyme starts as follows: As I was going to St. Ives I met a man with 7 wives.... Problem 1.7P: a. In how many ways can 3 boys and 3 girls sit in a row? b. In how many ways can 3 boys and 3 girls... Problem 1.8P: When all letters are used, how many different letter arrangements can be made from the letters a.... Problem 1.9P: A child has 12 blocks, of which 6 are black, 4 are red, 1 is white, and 1 is blue. If the child puts... Problem 1.10P: In how many ways can 8 people be seated in a row if a. there are no restrictions on the seating... Problem 1.11P: In how many ways can 3 novels. 2 mathematics books, and 1 chemistry book be arranged on a bookshelf... Problem 1.12P: How many 3 digit numbers zyz, with x, y, z all ranging from 0 to9 have at least 2 of their digits... Problem 1.13P: How many different letter permutations, of any length, can be made using the letters M 0 T T 0. (For... Problem 1.14P: Five separate awards (best scholarship, best leadership qualities, and so on) are to be presented to... Problem 1.15P: Consider a group of 20 people. If everyone shakes hands with everyone else, how many handshakes take... Problem 1.16P: How many 5-card poker hands are there? Problem 1.17P: A dance class consists of 22 students, of which 10 are women and 12 are men. If 5 men and 5 women... Problem 1.18P: A student has to sell 2 books from a collection of 6 math, 7 science, and 4 economics books. How... Problem 1.19P: Seven different gifts are to be distributed among 10 children. How many distinct results are... Problem 1.20P: A committee of 7, consisting of 2 Republicans, 2 Democrats, and 3 Independents, is to be chosen from... Problem 1.21P: From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. How... Problem 1.22P: A person has 8 friends, of whom S will be invited to a party. a. How many choices are there if 2 of... Problem 1.23P: Consider the grid of points shown at the top of the next column. Suppose that, starting at the point... Problem 1.24P: In Problem 23, how many different paths are there from A to B that go through the point circled in... Problem 1.25P: A psychology laboratory conducting dream research contains 3 rooms, with 2 beds in each room. If 3... Problem 1.26P: Show k=0n(nk)2k=3n Simplify k=0n(nk)xk Problem 1.27P: Expand (3x2+y)5. Problem 1.28P: The game of bridge is played by 4 players, each of w1om is dealt 13 cards. How many bridge deals are... Problem 1.29P: Expand (x1+2x2+3x3)4. Problem 1.30P: If 12 people are to be divided into 3 committees of respective sizes 3, 4, and 5, how many divisions... Problem 1.31P: If 8 new teachers are to be divided among 4 schools, how many divisions are possible? What if each... Problem 1.32P: Ten weight lifters are competing in a team weight-lifting contest. Of the lifters, 3 are from the... Problem 1.33P: Delegates from 10 countries, including Russia, France, England, and the United States, are to be... Problem 1.34P: If 8 identical blackboards are to be divided among 4 schools, how many divisions are possible? How... Problem 1.35P: An elevator starts at the basement with 8 people (not including the elevator operator) and... Problem 1.36P: We have 520.000 that must be invested among 4 possible opportunities. Each investment must be... Problem 1.37P: Suppose that 10 fish are caught at a lake that contains 5 distinct types of fish. a. How many... Problem 1.1TE: Prove the generalized version of the basic counting principle. Problem 1.2TE: Two experiments are to be performed. The first can result in any one of m possible outcomes. If the... Problem 1.3TE: In how many ways can r objects be selected from a set of n objects if the order of selection is... Problem 1.4TE: There are (nr) different linear arrangements of n balls of which r are black and nr are white. Give... Problem 1.5TE: Determine the number of vectors (x1,...,xn), such that each x1 is either 0 or 1 andi=1nxiK Problem 1.6TE: How many vectors x1,...,xk are there for which each xi is a positive integer such that1xin and... Problem 1.7TE: Give an analytic proof of Equation (4.1). 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![**Markov Chain Example**
Given the Markov Chain \((S, x_0, P)\) where \(S = \{s_1, s_2, s_3\}\), \(x_0 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\), and
\[ P = \begin{bmatrix}
0 & .3 & .2 \\
1 & 0 & .4 \\
0 & .7 & .4
\end{bmatrix} \]
\[P^2 = \begin{bmatrix}
.3 & .14 & .2 \\
0 & .58 & .36 \\
.7 & .28 & .44
\end{bmatrix} \]
\[P^3 = \begin{bmatrix}
.14 & .23 & .196 \\
.58 & .252 & .376 \\
.28 & .518 & .428
\end{bmatrix} \]
\[P^4 = \begin{bmatrix}
.23 & .1792 & .1984 \\
.252 & .4372 & .3692 \\
.518 & .3836 & .4344
\end{bmatrix} \]
\[P^5 = \begin{bmatrix}
.1792 & .20788 & .19704 \\
.4372 & .33264 & .37218 \\
.3836 & .45948 & .4308
\end{bmatrix} \]
\[P^6 = \begin{bmatrix}
.20788 & .191688 & .197808 \\
.33264 & .391672 & .36936 \\
.45948 & .41664 & .432832
\end{bmatrix} \]
Compute the probability that the Markov Chain is in state \(s_1\) at time 6.
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**Interpretation of Matrices:**
In this case, each matrix \(P^n\) represents the state transition probabilities after \(n\) steps.
For example:
- \(P^2\) indicates the state probabilities after 2 steps.
- \(P^3\) indicates the state probabilities after 3 steps.
- and so on...
As \(n\) increases, \(P^n\) gives the probabilities of being in each state after \(n\) transitions, starting](https://content.bartleby.com/qna-images/question/00a48d79-f805-418c-a679-1c91878d1d75/590b5f5e-52ef-4ed1-a14e-06092fbb209f/jwnjk4w_processed.png)
