Problem 45 - Applied Statistics and Probability for Engineers

Question

Consider the design of a communication system. (a) How many three-digit phone prefixes that are used to represent a particular geographic area (such as an area code) can be created from the digits 0 through \(9 ?\) (b) As in part (a), how many three-digit phone prefixes are possible that do not start with 0 or \(1,\) but contain 0 or 1 as the middle digit? (c) How many three-digit phone prefixes are possible in which no digit appears more than once in each prefix?

Step-by-Step Solution

Answer

(a) 1000 prefixes. (b) 160 prefixes. (c) 720 prefixes.

Step 1: Total Possible Phone Prefixes

To find the total number of three-digit phone prefixes, consider each digit separately. Each of the three digits can independently be any of the 10 digits from 0 to 9. Therefore, the number of possible combinations is calculated as a product: \[ 10 \times 10 \times 10 = 1000. \]

Step 2: Prefixes Starting with 2-9 and Middle Digit 0 or 1

For a prefix that does not start with 0 or 1, the first digit has 8 options (digits 2 through 9). The second digit must be either 0 or 1, providing 2 options. The third digit can be any digit from 0 to 9, resulting in 10 options. Multiply the numbers of choices for each digit:\[ 8 \times 2 \times 10 = 160. \]

Step 3: Unique Digits in Prefix

To have a prefix where each digit is different, start by picking the first digit (10 choices, as it can be any digit 0-9). The second digit must be different than the first, providing 9 remaining options. The third digit must be different from the first two, leaving 8 options. Therefore, the number of such prefixes is:\[ 10 \times 9 \times 8 = 720. \]

Key Concepts

Probability

Let's explore probability in the context of this exercise. Probability is the measure of the likelihood that an event will occur. In this case, it applies to how likely we are to randomly select a specific three-digit phone prefix from all possible combinations. To calculate probability, you divide the number of favorable outcomes (the combinations you are interested in) by the total number of possible outcomes.

For example, if you are interested in the probability of selecting a three-digit prefix that doesn't start with 0 or 1 and contains either 0 or 1 as the middle digit, you first find the total number of combinations (160 as calculated in the provided solution). Then you calculate the total number of possible prefixes (which is 1000). The probability would be the ratio of these two numbers: \[P(\text{Prefix condition}) = \frac{160}{1000} = 0.16.\]

This means there is a 16% chance of randomly selecting a prefix matching that condition out of all possible prefixes.

Permutations

Permutations refer to the different arrangements of a set of items, where the order of these items matters. In the context of our exercise, permutations are involved in generating different phone prefixes.

Consider the prefixes where no digit appears more than once. Each digit must be unique, requiring us to calculate permutations. We choose the first digit (10 choices), and for the second digit, we must pick from the remaining nine, reducing our options. For the third digit, we have 8 remaining choices. The formula for calculating this is:\[10 \times 9 \times 8 = 720\]

Therefore, there are 720 unique three-digit permutations where no digit repeats.

Digit Combinations

In combinatorics, combinations refer to grouping items where order does not matter. However, in this exercise, we are dealing with digit combinations in the sense of selecting specific sets of digits for phone prefixes.

The concept of digit combinations involves understanding how digits are selected based on given criteria. For instance, calculating how many three-digit combinations exist if they can start with any digit and include either 0 or 1 in the middle involves calculating different valid sequences.

• First digit: can be any of 2 through 9 (8 options).
• Second digit: must be 0 or 1 (2 options).
• Third digit: any from 0 to 9 (10 options).

These constraints lead us to calculate \[8 \times 2 \times 10 = 160\]

Indicating there are 160 possible combinations based on these criteria.

Mathematical Solutions

In solving exercises involving combinatorics, you often rely on structured processes to systematically break down complex problems. Mathematical solutions involve clear step-by-step computation, as shown in the original solution steps.

For calculating three-digit prefixes:

• **Step 1**: Determine total combinations by multiplying potential choices for each digit.
• **Step 2**: Apply conditions to narrow down options and use multiplication for combining choices (e.g., not starting with 0 or 1).
• **Step 3**: Ensure no repetition by systematically reducing choices for each subsequent digit (a permutation problem).

Each step is a strategic application of mathematical principles such as multiplication and permutation to achieve the desired outcome. Breaking problems down in this way enables clearer understanding and manageable solution paths.