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Chapter 16: Problem 40

\(\ln 38-43,\) solve for \(x\) $$ _{13} P_{5}=1,287\left(_{x} P_{x}\right) $$

Short Answer

Expert verified
The value of \(x\) is 5.

Step by step solution

01

Understand the Given

We're given the equation \[ _{13} P_{5} = 1,287 imes (_{x} P_{x}) \]where \(_n P_r\) denotes a permutation of \(r\) items out of \(n\).The task is to find the value of \(x\).
02

Calculate Known Permutation

Calculate \(_{13}P_{5}\) using the permutation formula:\[ _{n}P_{r} = \frac{n!}{(n-r)!} \]Thus,\[ _{13}P_{5} = \frac{13!}{(13-5)!} = \frac{13 \times 12 \times 11 \times 10 \times 9}{1} = 154440 \]
03

Simplify the Equation

Use the value of \(_{13} P_{5}\) in the equation:\[ 154440 = 1287 \times (_{x} P_{x}) \]To isolate \(_{x} P_{x}\), divide both sides by 1287:\[ _{x} P_{x} = \frac{154440}{1287} \]
04

Perform Division

Calculate the right-hand side:\[ _{x} P_{x} = 120 \]
05

Solve the Permutation Equation

Now, we need to find \(x\) such that:\[ _{x} P_{x} = x! = 120 \]Check values:- \(5! = 120\)Thus, \(x = 5\).

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Key Concepts

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

Factorials
Factorials are a fundamental concept in mathematics, especially when dealing with permutations and combinations. The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers up to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow quickly with larger numbers, which is why they are often used in calculations involving large sets of items.

A few points about factorials:
  • For \( n = 0 \), \( 0! \) is defined to be 1. This is an important base case in combinatorial mathematics.
  • Factorials are widely used not only in permutations but also in combinations and binomial theorem.
  • They simplify the calculation of permutations by reducing large numbers into a product of smaller, manageable factors.
Permutation formula
Permutations involve the arrangement of items where the order is important. The permutation formula is given by \( _{n}P_{r} = \frac{n!}{(n-r)!} \), where \( n \) is the total number of items, and \( r \) is the number of items to arrange. This formula calculates how many different ways we can arrange \( r \) items out of \( n \) without repetition.

Important aspects of permutations:
  • When \( r = n \), the formula simplifies to \( n! \) since you are arranging all available items.
  • Permutations differ from combinations. While permutations consider order, combinations do not.
  • Permutations are useful in scenarios like seating arrangements, ordering tasks, or determining race results.
In the problem, we used this formula to calculate \( _{13}P_{5} \), resulting in 154440, by arranging 5 items from a set of 13.
Solving equations
Solving equations is a common task where the goal is to find the values of variables that satisfy the equation. In these scenarios, each equation is like a puzzle, where mathematical techniques help you "piece together" the unknowns.

Approaches to solving equations include:
  • Isolation: Like in the given problem, we isolate the unknown by performing operations such as division, multiplication, addition, or subtraction on both sides.
  • Substitution: In more complex problems, we might substitute known values or expressions to simplify the equation.
  • Factorization: Breaking down expressions into products of simpler factors can make solving easier, especially for polynomial equations.
In the initial exercise, we simplified the permutation equation and solved \( _{x} P_{x} = 120 \). By matching it with its factorial equivalent \( 5! = 120 \), we identified \( x = 5 \). This straightforward solution relied on simple arithmetic operations and knowledge of factorials.

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