Problem 40 - Algebra 2 and Trigonometry

Question

A given region has the telephone prefix 472. a. How many 7-digit telephone numbers are possible with this prefix? b. How many of these possibilities end with 4 unique digits? c. How many of these possibilities form an even number?

Step-by-Step Solution

Answer

a. 10,000,000 numbers b. 5,040 numbers c. 5,000,000 numbers

Step 1: Calculate Total Possibilities for 7-digit Phone Numbers

To find the number of 7-digit telephone numbers possible, we look at the format: 472-XXXXXXX, where X represents any digit from 0 to 9. There are 7 digits, and each digit can be one of 10 choices. So, the total number of combinations is given by: \[ 10^7 \]. Evaluating this gives us \( 10,000,000 \) possible phone numbers.

Step 2: Determine Possibilities Ending with 4 Unique Digits

For telephone numbers ending with 4 unique digits, consider the format 472-XXXXXabcd where a, b, c, d are unique digits. The number of unique 4-digit sequences is calculated using permutations because each digit must be different: \( 10 \) options for 'a', \( 9 \) for 'b', \( 8 \) for 'c', and \( 7 \) for 'd'. Thus, the total number is: \[ 10 \times 9 \times 8 \times 7 = 5,040 \]. Since there are 3 other fixed digits, the total number remains 5,040.

Step 3: Calculate Possibilities that Form an Even Number

A number is even if it ends in 0, 2, 4, 6, or 8. In the format 472-XXXXXXd, where d is even, we have 5 choices for 'd'. For the first 6 digits, each can independently be chosen from 0-9, giving \( 10^6 \) combinations. Therefore, the total number is: \[ 10^6 \times 5 = 1,000,000 \times 5 = 5,000,000 \] possibilities that end in an even number.

Key Concepts

Permutations

Permutations are arrangements of items in a particular order. This makes permutations an important concept in combinatorics when considering problems that deal with the arrangement of distinct items. In the context of telephone numbers, permutations are particularly useful to determine how many ways unique digits can be arranged to form part of a phone number.

• Imagine you have to arrange 4 digits, each at a different position (e.g., last 4 digits of a phone number).
• For the first position, you have 10 options (digits 0-9).
• Once you've picked a digit, 9 choices remain for the second position, and so on.

Thus, the number of unique arrangements of 4 different digits is given by filling accessible spots with decreasing options: \[10 \times 9 \times 8 \times 7 = 5,040.\]This mathematical tool helps calculate all the potential combinations in scenarios needing ordered arrangements.

Even Numbers

Understanding even numbers is simple yet crucial when dealing with numeric sequences, like telephone numbers, that need to meet specific criteria. An even number is any integer that is divisible by 2 without leaving a remainder. In terms of digits, this means that the number ends with one of these: 0, 2, 4, 6, or 8.

• For a telephone number to be even, its last digit must be one of these five even digits.
• With each of the previous digits being 0-9, there are consistent options up apart from the last digit.

To compute all possible even telephone numbers, once the prefix is fixed (e.g., 472), you focus on choosing an even number for the last digit only. This provides you 5 options for that position, while the remaining positions are free to use any digit from 0 to 9. Hence, you compute: \[10^6 \times 5 = 1,000,000 \times 5 = 5,000,000.\]So, 5 million combinations of such numbers are possible when valid digits are taken into account.

Unique Digits

Unique digits in a combination ensure that no repetition occurs in the set of chosen digits. This restriction heavily influences how numbers are formed in tasks where digit repetition isn't allowed. For example, in deciding the last four digits of a phone number where each digit must be unique:

• The first digit has 10 possible values.
• After it is chosen, the second can be any of the remaining 9 digits.
• The process continues, reducing available options as you continue until all 4 unique digits are assigned.

This leads to permutations, calculating the possible arrangements without repetition: \[10 \times 9 \times 8 \times 7 = 5,040.\]The strategy towards securing unique digits integrates the principle of decreasing choices, ensuring all digit sequences are distinct at every step.

Telephone Numbers

Telephone numbers are often structured sequences of digits, typically designed to be easily memorable. Understanding the construction of these numbers, especially how different criteria affect the total combinations possible, is fundamental in combinatorics:

• They often have fixed parts, like prefixes (example: the 3-digit area code or prefix '472').
• The remaining digits must be arranged following the rules or constraints (such as ending in an even number).

Analyzing the given format 472-XXXXXXX, the totality of possible combinations arises from unrestricted digits (chosen freely from 0-9) and constraints requested in particular tasks. The calculation \[10^7 = 10,000,000\]represents the unfettered number of potential 7-digit numbers with this prefix without any other constraints. Constraining the number to ensure that all involved digits follow specific rules (uniqueness, parity) allows for various counting techniques to determine feasible options within each category.