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Chapter 9: Problem 7

Find the number. $$P(6,1)$$

Short Answer

Expert verified
There are 6 permutations of 6 items taken 1 at a time.

Step by step solution

01

Understand the notation

The notation \( P(n, r) \) refers to the number of permutations of \( n \) items taken \( r \) at a time. This means we are looking for arrangements of \( r \) items from a set of \( n \) items.
02

Apply the permutation formula

The formula for permutations is given by \( P(n, r) = \frac{n!}{(n-r)!} \). We need to plug in the values \( n = 6 \) and \( r = 1 \) into the formula.
03

Calculate the factorials

Calculate 6! and (6-1)!: \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \) and \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
04

Substitute into the formula

Substitute the calculated factorial values into the formula: \( P(6,1) = \frac{6!}{5!} = \frac{720}{120} \).
05

Simplify the expression

Simplify \( \frac{720}{120} \) to find the result of the permutation. Calculating this gives 6.

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Key Concepts

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

Factorial
The factorial of a number is a key component of understanding permutations. The concept "factorial" refers to the product of all positive integers up to that number.
  • It is denoted using an exclamation mark: so, \( n! \) means "n factorial".
  • For example, \( 5! \) would be calculated as \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
  • The factorial of 0 is defined to be 1, which can sometimes be surprising!
Factorials grow very quickly as the number increases. They are extremely useful in permutations and combinations, where they help us determine the number of different ways to arrange or select items.
Permutation Formula
The permutation formula allows us to determine the number of different ways we can arrange or order a set of items. To express it, we use \( P(n, r) \), where \( n \) is the total number of items and \( r \) is the number of items we want to arrange. The formula itself is:\[ P(n, r) = \frac{n!}{(n-r)!} \]
  • It essentially considers the total possible arrangements of \( n \) items and then removes the arrangements of \( n-r \) items鈥攖hose extra positions we do not need.
  • If \( r = n \), the formula simplifies to just \( n! \).
  • For smaller \( r \) values, like the exercise \( P(6, 1) \), it becomes very simple: it's just \( n \).
Using the permutation formula helps to solidify our understanding of different possible orderings of selections from a set.
Mathematical Notation
Mathematical notation provides a shorthand way to describe mathematical concepts and operations. In the context of permutations, the notation \( P(n, r) \) is particularly important. Here鈥檚 a bit more on mathematical notations:
  • "\( P \)" stands for "permutation" in this specific notation, indicating that we are calculating permutations.
  • The "\( n \)" and "\( r \)" parameters inside the parentheses tell us the total number of items to consider and how many we want to arrange.
  • Such notations help mathematicians and learners quickly communicate complex mathematical ideas without lengthy descriptions.
Understanding how to interpret and use mathematical notation is crucial for efficiently solving problems like permutation calculations and many other areas of mathematics.

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Most popular questions from this chapter

After selecting nine players for a baseball game, the manager of the team arranges the batting order so that the pitcher bats last and the best hitter bats third. In how many different ways can the remainder of the batting order be arranged?

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