/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 Evaluate each expression. $$P(... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 33

Evaluate each expression. $$P(99,0)$$

Short Answer

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Step by step solution

01

Understanding the Permutation Notation

Permutation notation, denoted as \(P(n, k)\), represents the number of ways to arrange \(k\) items out of \(n\) distinct items.
02

Set the Specific Values

In this case, we need to evaluate \(P(99, 0)\), where \(n = 99\) and \(k = 0\).
03

Apply the Permutation Formula

The formula for permutations is \(P(n, k) = \frac{n!}{(n - k)!}\). Substituting \(n = 99\) and \(k = 0\), we get: \[ P(99, 0) = \frac{99!}{(99 - 0)!} = \frac{99!}{99!} \]
04

Simplify the Expression

Since \(\frac{99!}{99!} = 1\), we conclude that \[ P(99, 0) = 1 \]

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Key Concepts

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

Permutation Formula
Permutations play a key role in understanding how we can arrange a set of items. The permutation formula is pivotal for calculating this. In essence, the permutation formula is: \[ P(n, k) = \frac{n!}{(n - k)!} \]This formula allows you to determine the number of ways to arrange \(k\) objects from a set of \(n\) distinct items. Here, \(n!\) (or \(n\) factorial) represents the product of all positive integers up to \(n\), and \((n - k)!\) represents the product of all positive integers up to \(n - k\). For example, if you want to find out how many ways you can arrange 2 items out of 4 distinct items, you would calculate \(P(4, 2)\) using the permutation formula.
Factorial
The factorial, denoted as \(n!\), is a central concept in permutations. It's the product of all positive integers up to a given number \(n\). Here are some examples to help you understand:
  • \(3! = 3 \times 2 \times 1 = 6 \)
  • \(4! = 4 \times 3 \times 2 \times 1 = 24 \)
  • \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
A special case is \(0!\), which is equal to 1 by definition.This might seem unusual at first, but it makes the math consistent and easier to work with. Factorials grow very quickly, that's why they are ideal for counting permutations among other things.
Permutation Notation
Permutation notation is a shorthand way of expressing a permutation scenario. It is written as \(P(n, k)\).
  • \(n\) represents the total number of distinct items.
  • \(k\) represents the number of items you are arranging from those \(n\) items.
So, if you see \(P(5, 3)\), it means you're looking at the number of ways to arrange 3 items out of a total of 5 distinct items. This notation simplifies the process of writing out and calculating permutations, making it easier to apply the permutation formula quickly.

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Most popular questions from this chapter

Use the addition rule to solve each problem. If \(P(A)=0.8, P(B)=0.6,\) and \(P(A \cap B)=0.5,\) then what is \(P(A \cup B) ?\)

Write the complete binomial expansion for each of the following powers of a binomial. $$(y-2)^{4}$$

Solve each problem. The probability that the Yankees win the World Series is \(9 / 10\), and the probability that the Yankees do not win the World Series is \(1 / 10 .\) What are the odds in favor of the Yankees winning the World Series?

Use the addition rule to prove that \(P(A)+P\left(A^{\prime}\right)=1\) for any complementary events \(A\) and \(A^{\prime}\)

Solve each problem. What is the coefficient of \(w^{2} x y^{3} z^{9}\) in the expansion of \((w+x+y+z)^{15} ?\)
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