Problem 33
Question
In Exercises 29 - 34, evaluate \( _nP_r \). \( _5P_4 \)
Step-by-Step Solution
Verified Answer
The value of the permutation \( _5P_4 \) is 120.
1Step 1: Understand the Formula for Permutations
The formula for permutations in this notation is: \( _nP_r = \frac{n!}{(n-r)!} \). Here, '!' denotes factorial which is the product of an integer and all the integers below it down to 1.
2Step 2: Plug the values into the formula
Now that we understand the formula, we can simply plug in our values for \( n \) and \( r \) into it. Doing that gives us \(_5P_4 = \frac{5!}{(5-4)!} \).
3Step 3: Solve for Factorials and Evaluate the Expression
Next, we need to compute the factorials. The factorial of 5 is \(5! = 5 * 4 * 3 * 2 * 1 = 120\), and the factorial of 1 (because \(5 - 4 = 1\)) is \(1! = 1\). Now we substitute these results into the equation to get \( _5P_4 = \frac{120}{1} = 120\).
Key Concepts
Factorial NotationCombinatoricsPermutation Formula
Factorial Notation
When we talk about factorial notation in mathematics, we're referring to a very specific operation that is central to many areas of combinatorics and probability. A factorial is represented by the exclamation mark (!) and is defined for a non-negative integer, say, n. The factorial of n, denoted as n!, is the product of all positive integers less than or equal to n. For example,
4! = 4 × 3 × 2 × 1 = 24.
It's important to note that the factorial of zero is defined to be 1, that is,
0! = 1.
Understanding factorial notation is key to solving problems involving permutations and combinations, as these concepts often require manipulating factorials to arrive at a solution.
4! = 4 × 3 × 2 × 1 = 24.
It's important to note that the factorial of zero is defined to be 1, that is,
0! = 1.
Understanding factorial notation is key to solving problems involving permutations and combinations, as these concepts often require manipulating factorials to arrive at a solution.
Combinatorics
Combinatorics is a fascinating and far-reaching field of mathematics that deals with counting, both as an end in itself and as a means to analyze and solve more complex problems. It includes the study of combinations, permutations, and various structures such as graphs and lattices. In combinatorics, permutations refer to the different ways of arranging a set of items where the order is important. Conversely, combinations are about grouping items where the order doesn't matter.
For students, getting comfortable with the basic principles of combinatorics opens the door to understanding more intricate mathematical concepts and enhances their problem-solving skills. This field of mathematics is not only theoretical; it has practical applications in many areas such as computer science, optimization, and statistical physics.
For students, getting comfortable with the basic principles of combinatorics opens the door to understanding more intricate mathematical concepts and enhances their problem-solving skills. This field of mathematics is not only theoretical; it has practical applications in many areas such as computer science, optimization, and statistical physics.
Permutation Formula
The permutation formula is a specific application of combinatorial mathematics. It allows us to calculate the number of ways to arrange a subset of a set when the order of arrangement is important. The general formula for permutations is given by
\( _nP_r = \frac{n!}{(n-r)!} \),
where n is the total number of items to choose from, r is the number of items being chosen, n! is the factorial of n, and (n-r)! is the factorial of the difference between n and r. This formula is derived from the principle that each choice has fewer options than the previous one.
For instance, if you have 5 books and want to know in how many ways you can arrange 4 of them on a shelf, you would use the permutation formula to find that there are
\( _5P_4 = 120 \)
ways to do so. The sequence of arrangements matters here, which is why permutations, rather than combinations, are the appropriate calculation.
\( _nP_r = \frac{n!}{(n-r)!} \),
where n is the total number of items to choose from, r is the number of items being chosen, n! is the factorial of n, and (n-r)! is the factorial of the difference between n and r. This formula is derived from the principle that each choice has fewer options than the previous one.
For instance, if you have 5 books and want to know in how many ways you can arrange 4 of them on a shelf, you would use the permutation formula to find that there are
\( _5P_4 = 120 \)
ways to do so. The sequence of arrangements matters here, which is why permutations, rather than combinations, are the appropriate calculation.
Other exercises in this chapter
Problem 32
In Exercises 9-32, write the first five terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = \dfrac{(-1)^{n + 1}}{2n + 1} \)
View solution Problem 33
In Exercises 31 - 34, find the probability for the experiment of drawing two marbles (without replacement) from a bag containing one green, two yellow, and thre
View solution Problem 33
In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(x^2 + y^2\right)^4 \)
View solution Problem 33
In Exercises 31 - 42, use mathematical induction to prove the property for all positive integers \( n \). If \( x_1 \neq 0, x_2 \neq 0, , \cdots , x_n \neq 0 \)
View solution