Problem 77 - College Algebra

Question

Write a word problem that can be solved by evaluating \(_7 P_{3}\)

Step-by-Step Solution

Answer

Hence, there are 210 different ways to distribute the trophies among the players.

Step 1: Understand the problem in terms of permutation

Here is an example of a problem that we can solve with \(_7 P_{3}\): Suppose there are 7 players in a tennis tournament. The top 3 places (1st, 2nd and 3rd) will receive trophies. How many different ways can the trophies be distributed among the players?

Step 2: Applying the permutation formula

We can represent this problem with the permutation formula, looking for \(_7 P_{3}\) which will tell us the different ways to distribute 3 trophies among 7 players, where order matters. The formula for permutation is: \(_n P_k = \frac{n!}{(n-k)!}\) where \(n!\) is the factorial of n, meaning a product of an integer and all the integers below it, and \(k\) is the number of objects we are choosing from n.

Step 3: Calculate the permutation

Applying the formula for our specific problem, where \(n = 7\) and \(k = 3\), the calculation becomes \(_7 P_3 = \frac{7!}{(7-3)!} = \frac{7*6*5*4*3*2*1}{4*3*2*1} = 7*6*5 = 210\)

Key Concepts

Factorial Notation

Factorial notation is essential in various mathematical concepts, especially in permutations and combinations. In factorial notation, the factorial of a non-negative integer 'n', denoted as 'n!', is the product of all positive integers less than or equal to 'n'. For instance, \[\begin{equation}5! = 5 \times 4 \times 3 \times 2 \times 1 = 120.\end{equation}\]The value of '0!' is defined to be 1, which is an important convention in combinatorics. Factorial notation is particularly useful because it compactly represents the total number of ways to arrange a certain number of objects.Moreover, factorial notation is used to calculate permutations, which are specific arrangements of objects where the order is important. Understanding how to work with factorials is crucial when tackling problems involving permutations, as we'll see in the application of the permutation formula.

Permutation Formula

The permutation formula helps determine the number of possible arrangements of a subset of items within a larger set where the order of arrangement matters. The standard notation for a permutation is \[\begin{equation}_nP_k, \end{equation}\]which represents the number of ways to arrange 'k' items from a set of 'n' items. The formula for permutations is \[\begin{equation}_nP_k = \frac{n!}{(n-k)!}.\end{equation}\]In this context, '!'

Combinatorics

Combinatorics is a field of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is related to many other areas of mathematics, such as algebra, probability, and geometry, and it is often used in real-world applications in computer science, statistics, and physics.One of the key concepts in combinatorics is permutations, which is a counting technique to determine the number of possible arrangements of a set where the order of selection is important. Permutations are contrasted with combinations, where the order does not matter.Understanding combinatorics is pivotal when solving problems related to probability, as it allows for the precise calculation of likelihoods of specific events. Learning how to manipulate combinatorial expressions is thus invaluable for students interested in a wide range of mathematical and practical applications.