Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Classify each problem according to whether it involves a permutation or a combination. As part of a quality-control program, 3 cell phones are selected at random for testing from 100 cell phones produced by the manufacturer. In how many ways can this test batch be chosen?

Step-by-Step Solution

Answer

This problem involves a combination since the order of selected cell phones for testing does not matter. To find the number of ways to choose 3 cell phones from 100, we can use the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) with n = 100 and r = 3. Calculating, we get \(\binom{100}{3} = \frac{100!}{3!(100-3)!} = \frac{100\times99\times98}{3\times2\times1}=161700\). Therefore, there are 161,700 ways to choose the test batch of 3 cell phones from the produced 100 cell phones.

Step 1: Identify the values of n and r

In this problem, n represents the total number of cell phones, and r represents the number of cell phones we want to select for testing. We have 100 cell phones in total, and we want to select 3 of them. So, n = 100 and r = 3.

Step 2: Calculate the combination using the formula

Using the combination formula and the values of n and r, we can calculate the number of ways to choose 3 cell phones from 100 cell phones: \(\binom{100}{3} = \frac{100!}{3!(100-3)!}\)

Step 3: Simplify the expression

Simplify the expression in the numerator and the denominator: \(\binom{100}{3} = \frac{100!}{3!97!}\)

Step 4: Calculate the value

Now, we can calculate the value of the combination: \(\binom{100}{3} = \frac{100\times99\times98}{3\times2\times1}=161700\) So, there are 161,700 ways to choose the test batch of 3 cell phones from the 100 cell phones produced by the manufacturer.

Key Concepts

Understanding Permutations

Permutations are an essential concept in combinatorics where the order of selection matters. In simpler terms, permutations are arrangements of items in a specific sequence. Consider when you are assigning prizes to a group of winners, it would matter if one person receives the first prize instead of the second. This is a typical case where permutations come into play.

To calculate permutations, the formula is \[ P(n, r) = \frac{n!}{(n-r)!} \]. Here, \( n \) is the total number of items to choose from, and \( r \) is the number of items being chosen. This formula helps in determining all possible arrangements when picking \( r \) items out of \( n \).

Examples where permutations are necessary include seating arrangements, races, and list rankings where the order or position of choices is important. Permutations give us insight into the different ways to arrange or sequence these selections.

The Role of Combinations

Combinations are another crucial part of combinatorics, differing from permutations by ignoring the order of selection. In simpler scenes like choosing players for a sports team, it doesn't matter who is picked first or last, as long as the team is formed. Such scenarios where order doesn't matter involve combinations.

The formula for calculating combinations is\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]. Here, \( n \) represents the total number of items, and \( r \) stands for the number of items to choose. This formula helps determine the total number of ways \( r \) items can be selected from \( n \) without concern for order.

Examples for combination scenarios might involve selecting members for committees, creating teams, or drawing lottery numbers. Combinations allow us to understand the possible groupings without the emphasis on their sequences.

Factorial Mathematics Explained

Factorial mathematics is a fundamental concept used in both permutations and combinations. A factorial, denoted with an exclamation mark (\(!\)), is the product of all positive integers up to a specified number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). The factorial function grows rapidly with larger integers.

It provides a way to compute the total number of ways to arrange a set of items completely. Factorials are critical in simplifying the expressions in permutation and combination calculations by breaking them down into manageable multiplications.

By using factorials, we can efficiently solve complex combinatorics problems by either substituting them directly into formulas or simplifying them in equations. Common uses of factorials are in computing probabilities, optimizing algorithms, and analyzing data.