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Chapter 5: Problem 12

Find the value of each permutation. $$_7 P_{2}$$

Short Answer

Expert verified
The value of \( _7P_{2} \) is 42.

Step by step solution

01

Understand the Permutation Formula

The formula for permutations is given by \[ P(n, k) = \frac{n!}{(n-k)!} \] where \(n\) is the total number of items, and \(k\) is the number of items to choose.
02

Identify the Values

In the problem, \(n = 7\) and \(k = 2\). So, we need to find \[ P(7, 2) = \frac{7!}{(7-2)!} \]
03

Calculate Factorials

Calculate the factorials: \[ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 \] and \[ (7-2)! = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
04

Substitute and Simplify

Substitute the factorials into the permutation formula: \[ P(7, 2) = \frac{5040}{120} \] Simplify the fraction: \[ \frac{5040}{120} = 42 \]

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Key Concepts

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

Factorials
Factorials are important in math, especially in permutations and combinations. A factorial of a positive integer, denoted by an exclamation point (!), is the product of all positive integers up to that number. For example, the factorial of 5, written as 5!, is \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]
Factorials grow very quickly with even small increases in the number.
  • 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
  • 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040
Factorials are used to calculate permutations, which we will explore next.
Combinatorics
Combinatorics is the branch of mathematics focused on counting, arrangement, and combination of objects. It includes the study of permutations and combinations. When we discuss permutations, we refer to the different ways to arrange a set of objects. In combinatorics, the order of objects matters in permutations, unlike combinations.

For instance, arranging the numbers 1, 2, and 3 in different orders yields:
  • 123
  • 132
  • 213
  • 231
  • 312
  • 321
These are all different permutations. In contrast, combinations disregard the order, focusing only on the selection of items.
Permutation Formula
Permutations involve the arrangement of objects where the order is important. The formula to find the number of permutations when selecting 'k' items from 'n' items is: \[P(n, k) = \frac{n!}{(n-k)!}\] In our example, we want to find the number of ways to arrange 2 items out of 7, so we use the formula: \[P(7, 2) = \frac{7!}{(7-2)!}\] First, calculate the factorials: \[7! = 5040\] and \[5! = 120\] Then, substitute these values back into the formula: \[P(7, 2) = \frac{5040}{120} = 42\] Therefore, there are 42 different ways to arrange 2 items out of 7.

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Most popular questions from this chapter

A gene is composed of two alleles. An allele can be either dominant or recessive. Suppose a husband and wife, who are both carriers of the sickle- cell anemia allele but do not have the disease, decide to have a child. Because both parents are carriers of the disease, each has one dominant normal-cell allele and one recessive sickle-cell allele. Therefore, the genotype of each parent is \(S s .\) Each parent contributes one allele to his or her offspring, with each allele being equally likely. (a) List the possible genotypes of their offspring. (b) What is the probability that the offspring will have sickle-cell anemia? In other words, what is the probability the offspring will have genotype \(s s ?\) Interpret this probability. (c) What is the probability that the offspring will not have sickle-cell anemia but will be a carrier? In other words, what is the probability that the offspring will have one dominant normal-cell allele and one recessive sickle- cell allele? Interpret this probability.

Find the value of each combination. $$_{12} C_{3}$$

Four members from a 20 -person committee are to be selected randomly to serve as chairperson, vice-chairperson, secretary, and treasurer. The first person selected is the chairperson; the second, the vice-chairperson; the third, the secretary; and the fourth, the treasurer. How many different leadership structures are possible?

Determine the probability that at least 2 people in a room of 10 people share the same birthday, ignoring leap years and assuming each birthday is equally likely by answering the following questions: (a) Compute the probability that 10 people have different birthdays. (Hint: The first person's birthday can occur 365 ways; the second person's birthday can occur 364 ways, because he or she cannot have the same birthday as the first person; the third person's birthday can occur 363 ways, because he or she cannot have the same birthday as the first or second person; and so on.) (b) The complement of "10 people have different birthdays" is "at least 2 share a birthday." Use this information to compute the probability that at least 2 people out of 10 share the same birthday.

Find the value of each combination. $$_{9} C_{2}$$
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