Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

In how many ways can a quality-control engineer select a sample of 3 microprocessors for testing from a batch of 100 microprocessors?

Step-by-Step Solution

Answer

Using the combination formula, a quality-control engineer can select a sample of 3 microprocessors for testing from a batch of 100 microprocessors in \(C(100, 3) = \frac{100!}{3!(97)!} = 161,700\) ways.

Step 1: Understand the combination formula

The combination formula is used to find the number of ways to choose k items from a set of n items without considering the order and without replacement. The formula is: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where n is the total number of items, k is the number of items to choose, and ! denotes the factorial of a number (e.g., 5! = 5 × 4 × 3 × 2 × 1).

Step 2: Apply the combination formula to the problem

In this problem, we have a batch of 100 microprocessors (n = 100) and we want to select a sample of 3 for testing (k = 3). So, by applying the combination formula, we get: \[ C(100, 3) = \frac{100!}{3!(100-3)!} \]

Step 3: Compute and simplify the result

Now, we just need to perform the calculations and simplify the expression: \[ C(100, 3) = \frac{100!}{3!(97)!} \] Since we are calculating the factorial, we can cancel out part of the values to simplify the expression: \[ C(100, 3) = \frac{100 × 99 × 98 × 97!}{3! × 97!} \] The 97! terms cancel out: \[ C(100, 3) = \frac{100 × 99 × 98}{3!} \] Now, we calculate 3!: \[ C(100, 3) = \frac{100 × 99 × 98}{(3 × 2 × 1)} \] Finally, we perform the calculation: \[ C(100, 3) = \frac{100 × 99 × 98}{6} = 161700 \] So there are 161,700 ways for the quality-control engineer to select a sample of 3 microprocessors for testing from a batch of 100 microprocessors.

Key Concepts

Combination Formula

The combination formula is a fundamental concept in combinatorics, used to determine how many ways you can choose a subset of items from a larger set without considering the order of selection. Imagine you have a group of 100 microprocessors and you need to select 3 of them for testing while not caring about the arrangement in which they are chosen. The combination formula helps solve such problems efficiently by not overcounting different arrangements of the same items. The formula is given as:\[C(n, k) = \frac{n!}{k!(n-k)!}\]Where:

• \(n\) is the total number of items.
• \(k\) is the number of items to choose.
• \(!\) represents a factorial, which is the product of all positive integers up to that number.

This formula assumes no items are repeated, and typically you apply it in contexts where the order does not matter. Understanding and applying this concept is crucial in planning strategies for various scenarios, from quality control to organizing events.

Factorial Calculations

Factorials are an essential component of the combination formula, represented by the exclamation mark (!) notation. Calculating factorials involves multiplying a series of descending natural numbers.For example, the expression \(5!\) means you multiply 5 by every positive integer less than itself:\(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)Factorials grow very fast, making them useful for solving problems that involve large sets. In combinatorial formulas, like combinations, factorials define the number of potential ways to order a set of items. This is crucial for canceling terms and simplifying the calculations.In the specific exercise involving 100 microprocessors, you use 3 factorial \(3!\) when calculating combinations:\(3! = 3 \times 2 \times 1 = 6\)By understanding factorials, you'll better appreciate why certain terms cancel out when simplifying expressions, making it easier to arrive at the correct result.

Sample Selection

When you need to choose a sample from a larger set for testing or analysis, the concept of sample selection comes into play. It involves selecting a subset from a bigger group while considering various constraints such as randomness and replacing or not replacing items.In statistical and quality control contexts, like the scenario of selecting microprocessors for testing, you aim to balance the accuracy and reliability of the results with the feasibility of testing every item. This becomes significant when dealing with large batches, such as 100 microprocessors, and choosing just 3 for analysis.When applying the combination formula, you assume that each item is equally likely to be chosen, ensuring a fair and unbiased selection process. The formula, \(C(100, 3)\), calculates every possible group of 3 microprocessors you can choose from the 100 without concerning order.While straightforward in calculation, the idea of selecting a sample this way ensures that decisions are statistically valid and sound, aiding quality-control engineers in making informed choices about which items to test.