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Chapter 5: Problem 17

Find the value of each permutation. $$ { }_{8} P_{3} $$

Short Answer

Expert verified
336

Step by step solution

01

Understand the notation

{ }_{n} P_{r} represents the number of permutations of n items taken r at a time. The formula for permutations is given by: \[ { }_{n} P_{r} = \frac{n!}{(n-r)!} \]
02

Identify the values

In this problem, n = 8 and r = 3. We need to substitute these values into the permutation formula.
03

Calculate the factorials

Calculate the factorial values: \[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \]and \[ (8-3)! = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
04

Substitute and simplify

Substitute the factorial values into the permutation formula: \[ { }_{8} P_{3} = \frac{8!}{5!} = \frac{40320}{120} \]which simplifies to: \[ { }_{8} P_{3} = 336 \]

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Key Concepts

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

factorials
In math, a factorial is a number multiplied by all the positive integers less than it. It's represented by an exclamation mark (!). So, for any positive integer n, the factorial of n, written as n!, is calculated as: \( n! = n \times (n-1) \times (n-2) \times \text{...} \times 1 \) Factorials grow very fast. For instance, 5! equals: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \) Moreover, there are a few useful properties of factorials:
  • 0! is defined as 1.
  • Factorials can be used in combinations and permutations.
permutation formula
The permutation formula helps determine the number of ways to arrange a subset of items from a larger set, where the order of arrangement matters. The formula for permutations is represented as: \( {}_{n}P_{r} = \frac{n!}{(n-r)!} \) Here, n is the total number of items, and r is the number of items to arrange. Let's break down the formula:
  • n!: Factorial of the total items.
  • (n-r)!: Factorial of the difference between the total items and the items to arrange.
By dividing n! by (n-r)!, we eliminate the unnecessary part of the product and get the correct number of permutations.
calculating permutations
To concretely calculate permutations, let's use the problem: \( {}_{8}P_{3} \) We follow these steps:
  • Identify n and r values: Here, n = 8, and r = 3.
  • Use the permutation formula: \( {}_{n}P_{r} = \frac{n!}{(n-r)!} \)
  • Calculate the factorials: \( 8! = 40320 \) and \( 5! = 120 \), because \( (8-3)! = 5! \)
  • Substitute and simplify: \( {}_{8}P_{3} = \frac{8!}{5!} = \frac{40320}{120} = 336 \).
Hence, the number of ways to arrange 3 items out of 8 is 336.

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