Problem 81
Question
In Exercises 79 - 86, solve for \( n \). \( _nP_4 = 10 \cdot _{n - 1} P_3 \)
Step-by-Step Solution
Verified Answer
The solution to the equation is \( n = 5 \)
1Step 1: Understand the permutation
A Permutation is the arrangement of objects where the order matters. In mathematics, It's represented as \( _nP_k \) which can also be written as \( n! / (n - k)! \). Here, \( n! \) means factorial of n (product of all positive integers up to n).
2Step 2: Convert the given permutation equation into factorial
The given permutation is \( _nP_4 = 10 * _{n - 1} P_3 \). we need to convert it into factorials form which leads to \( \frac{n!}{(n - 4)!} = 10 * \frac{(n - 1)!}{(n - 4)!} \)
3Step 3: Simplifying the expression
Dividing the left and right sides of the equation by \( (n - 4)! \), they cancel out. This simplifies the equation to \( n! = 10(n -1)! \). This simplification allows the problem to be solved directly for \( n \).
4Step 4: Find the value of n
The value of n can be found by expressing both \( n! \) and \( (n-1)! \) in terms of factorials and equating the expressions. Hence, if n = 5, then the equation balances, as \( 5! = 10*(4!) \), or \( 120 = 10*24 \)
Key Concepts
Factorial NotationArrangement of Objects in MathematicsSolving Permutations
Factorial Notation
Factorial notation is a mathematical concept used to denote the product of a series of descending natural numbers. It is represented by an exclamation point (!). For any non-negative integer, say, n, the factorial is written as and calculated through the expression:
\( n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1 \)
For example, the factorial of 5 (denoted as 5!) would be \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). One particularly interesting trait of factorial is that \(0!\) is defined to be 1, despite the fact that 0 is not a positive number. This convention ensures the proper behavior of combinations and permutations when no items are selected.
Understanding factorial notation is essential for solving permutations, as they frequently require factorial computations to determine the number of possible arrangements.
\( n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1 \)
For example, the factorial of 5 (denoted as 5!) would be \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \). One particularly interesting trait of factorial is that \(0!\) is defined to be 1, despite the fact that 0 is not a positive number. This convention ensures the proper behavior of combinations and permutations when no items are selected.
Understanding factorial notation is essential for solving permutations, as they frequently require factorial computations to determine the number of possible arrangements.
Arrangement of Objects in Mathematics
The arrangement of objects in mathematics is a fundamental concept often dealt with in combinatorics, which is a field of study concerned with counting, both as a means and an end in obtaining results. When we talk about the arrangement of objects, we distinguish between two core ideas - permutations and combinations.
In permutations, the order of the objects matters. This can be seen in scenarios such as lining up students for a photo, where Peter, John, and Mary standing in a different sequence represents a different permutation, even though the group of students is the same.
\( _{10}P_3 = \frac{10!}{(10 - 3)!} \)
Where the notation \( _{n}P_{k} \) denotes the permutation of n objects taken k at a time. It is an important concept in probability theory as well, where ordering outcomes is vital.
In permutations, the order of the objects matters. This can be seen in scenarios such as lining up students for a photo, where Peter, John, and Mary standing in a different sequence represents a different permutation, even though the group of students is the same.
Understanding Permutations
In the context of permutations, we refer to the number of ways to order a subset of a set. For instance, by using permutation notation, we can calculate the possible arrangements of 3 books picked from a shelf of 10 books. The formula to calculate this is given by:\( _{10}P_3 = \frac{10!}{(10 - 3)!} \)
Where the notation \( _{n}P_{k} \) denotes the permutation of n objects taken k at a time. It is an important concept in probability theory as well, where ordering outcomes is vital.
Solving Permutations
Solving permutations involves finding the number of ways in which you can arrange a set or a subset of items. The general approach relies on the formula for permutations, which is expressed as \( _nP_k = \frac{n!}{(n-k)!} \). This equation succinctly represents the factorial notation and allows us to solve for the number of arrangements of k objects from a set of n objects.
In the context of our exercise example, to solve for \( _nP_4 = 10 \times _{n - 1}P_3 \), we applied the permutation formula to replace \( _nP_k \) with factorial expressions. By setting the rearranged factorial equation equal to the given value, we isolate 'n' and find the numerical solution.
It's through these principles and techniques that permutations are solved, allowing us to tackle a diverse array of problems about order and arrangement in mathematical contexts.
In the context of our exercise example, to solve for \( _nP_4 = 10 \times _{n - 1}P_3 \), we applied the permutation formula to replace \( _nP_k \) with factorial expressions. By setting the rearranged factorial equation equal to the given value, we isolate 'n' and find the numerical solution.
Simplification Techniques
A powerful simplification technique for solving permutation equations is to cancel out like terms. For instance, when we divide both sides by \((n - 4)!\), factorials common to both sides of the equation are eliminated, simplifying our problem to an easier, solvable equation.It's through these principles and techniques that permutations are solved, allowing us to tackle a diverse array of problems about order and arrangement in mathematical contexts.
Other exercises in this chapter
Problem 80
In Exercises 79 - 82, use a graphing utility to graph the first \( 10 \) terms of the sequence. (Assume \( n \) that begins with \( 1 \).) \( a_n = -5 + 2n \)
View solution Problem 80
In Exercises 77-84, simplify the factorial expression. \( \dfrac{10! \cdot 3!}{4! \cdot 6!} \)
View solution Problem 81
In Exercises 79 - 82, use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 79, use the expansion \( \
View solution Problem 81
In Exercises 81 - 85, determine whether the statement is true or false. Justify your answer. If the statement \( p_1 \) is true but the true statement \( P_6 \)
View solution