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Chapter 4: Problem 22

Express all probabilities as fractions. A classic counting problem is to determine the number of different ways that the letters of "Mississippi" can be arranged. Find that number.

Short Answer

Expert verified
There are 34,560 ways to arrange the letters in 'Mississippi'.

Step by step solution

01

Understand the problem

To find the number of different ways to arrange the letters in 'Mississippi', consider that some letters are repeated.
02

Identify unique letters and their frequencies

List the letters in 'Mississippi' and count their occurrences: 'M' appears 1 time, 'i' appears 4 times, 's' appears 4 times, and 'p' appears 2 times.
03

Use the formula for permutations with repetitions

The formula to find the number of distinct permutations of a word with repeated letters is: \[ \frac{n!}{n_1! \times n_2! \times \text{...} \times n_k! } \] where \(n\) is the total number of letters, and \(n_1, n_2, \text{...}, n_k\) are the frequencies of the repeated letters.
04

Substitute the values

In 'Mississippi', there are 11 letters in total, with the following frequencies: 1 ('M'), 4 ('i'), 4 ('s'), and 2 ('p'). Substitute these into the formula: \[ \frac{11!}{1! \times 4! \times 4! \times 2!} \]
05

Calculate the factorials

Calculate each factorial: \(11! = 39916800\), \(1! = 1\), \(4! = 24\), and \(2! = 2\).
06

Compute the final answer

Plug the factorials into the formula and simplify: \[ \frac{39916800}{1 \times 24 \times 24 \times 2} = \frac{39916800}{1152} = 34560 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Counting Problems
Counting problems are fundamental in probability and combinatorics. They involve determining the number of possible outcomes in different scenarios. Understanding counting problems is crucial for solving complex mathematical puzzles and real-world applications. When working with counting problems, it's essential to identify if the problem involves permutations, combinations, or other counting principles. In this case, we deal with permutations because the order of the letters matters. Identifying repetitions in the elements is also important because it impacts how we compute the total number of arrangements.
Permutations with Repetitions
Permutations are arrangements of elements, where the order matters. When elements are repeated, we use the permutations with repetitions formula to avoid overcounting. The formula is: \[ \frac{n!}{n_1! \times n_2! \times \text{...} \times n_k! } \] Here, \(n\) is the total number of elements, and \(n_1, n_2, ..., n_k\) are the counts of each unique element. For example, in 'Mississippi,' \(n = 11\) (total letters), \(n_1 = 1\) for 'M,' \(n_2 = 4\) for 'i,' \(n_3 = 4\) for 's,' and \(n_4 = 2\) for 'p.' Substituting these into the formula gives the number of distinct permutations for the word.
Factorials
Factorials are a key concept in permutations and counting problems. Denoted by \(!\), a factorial of a non-negative integer \(n\) is the product of all positive integers less than or equal to \(n\). For example: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] In 'Mississippi,' the calculation involves: \[ 11! = 39916800, \ 1! = 1, \ 4! = 24, \text{ and } 2! = 2 \] Plugging these into the permutations with repetitions formula: \[ \frac{11!}{1! \times 4! \times 4! \times 2!} = \frac{39916800}{1 \times 24 \times 24 \times 2} = \frac{39916800}{1152} = 34560 \] This result shows the total unique arrangements of the letters in 'Mississippi.'

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Most popular questions from this chapter

Find the probability.At Least One. In Exercises \(5-12,\) find the probability. Based on a poll conducted through the e-edition of USA Today, 67\% of Internet users are more careful about personal information when using a public Wi-Fi hotspot. What is the probability that among four randomly selected Internet users, at least one is more careful about personal information when using a public Wi-Fi hotspot? How is the result affected by the additional information that the survey subjects volunteered to respond?

Express all probabilities as fractions. Mendel conducted some his famous experiments with peas that were either smooth yellow plants or wrinkly green plants. If four peas are randomly selected from a batch consisting of four smooth yellow plants and four wrinkly green plants, find the probability that the four selected peas are of the same type.

Same Birthdays If 25 people are randomly selected, find the probability that no 2 of them have the same birthday. Ignore leap years.

Express all probabilities as fractions. A Social Security number consists of nine digits in a particular order, and repetition of digits is allowed. After seeing the last four digits printed on a receipt, if you randomly select the other digits, what is the probability of getting the correct Social Security number of the person who was given the receipt?

Answer the given questions that involve odds. In the Kentucky Pick 4 lottery, you can place a "straight" bet of \(\$ 1 dollars by selecting the exact order of four digits between 0 and 9 inclusive (with repetition allowed), so the probability of winning is \)1 / 10,000 .$ If the same four numbers are drawn in the same order, you collect \$5000, so your net profit is \$4999. a. Find the actual odds against winning. b. Find the payoff odds. c. The website www.kylottery.com indicates odds of 1: 10,000 for this bet. Is that description accurate?
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